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# After 2018 Bolsonaro victory, is a 2022 remake feasible?

Nuno Crokidakis 1 Serge Galam 2 1 Instituto de Física, Universidade Federal
Fluminense, Niterói, Rio de Janeiro, Brazil
2 CEVIPOF - Centre for Political Research, Sciences Po and CNRS, Paris, France

###### Abstract

We propose a contagion model to describe the evolution of the political voting
trends in Brazil after the dictatorship from 1985 to nowadays. We consider a
fully-connected population divided in two voting groups, left and right. Each
group includes three kinds of agents, sensitives, inflexibles and radicals.
While sensitives may shift their left or right voting, inflexibles and
radicals do not. Excluding political interactions with radicals, the model has
one interaction parameter, and we found the values which reproduce all the
voting outcomes of past presidential elections from 1989 till 2018. As an
alternative approach to explain 2018 election, we found that adding an empty
third voting group overlapping left and right, the populist voting group, can
also yield Bolsonaro 2018 victory. The initial filling of the populist group
is triggered by the breaking of the interaction barrier between sensitives and
radicals, which had prevailed for decades. Within each voting group, some of
the interacting pairs sensitive/radical recast the sensitive into a populist.
This status change from sensitive to populist allows the populist to interact
with sensitives of both left and right, thus creating an additional source of
filling of the populist group. The process is shown to yield the 2018 $55\%$
for Bolsonaro. From the 2018 distribution of votes, we evaluate the parameter
changes which recover the recent 2021 poll yielding a victory of Lula against
Bolsonaro at a score of 64/36. Thus, we compare the results for the two
scenarios, with and without populists, for 2018 election and 2021 poll.
Regarding the two possible scenarios for the 2018 Bolsonaro victory, we
discuss about a hypothetic Lula/Bolsonaro 2022 second round voting and the
requirements to yield either Lula or Bolsonaro victory in 2022. Both scenarios
are feasible.

Dynamics of social systems, Collective phenomena, Opinion Dynamics, Populism

##  I Introduction

The rise of populism has been observed in many democratic countries in the
last years [1, 2, 3]. Two most emblematic cases are the elections of Donald
Trump in the 2016 US election [4] and Jair Bolsonaro in the 2018 Brazilian
election [5]. In parallel to those successful victories of populist leaders,
far-right has grown in a series of Western European countries including
France, Germany, Italy and Spain [6].

Most of these drastic changes in electoral landscapes were not anticipated by
polls and pandits, taken them by surprise. The underlying mechanisms driving
the opinion dynamics towards populism and its connection to extreme political
views on both the right and the left are still not well understood. In
particular the various amplitudes from one country to another are distributed
over quite a large spectrum of values.

A recent study investigated the issue in Germany analyzing the changes in
political narratives [7]. Here we tackle the problem from the viewpoint of
opinion dynamics studied within the frame of sociophysics. A great deal of
works have been published along this approach [8, 9].

Several studies related to politics and elections were published in the last
years [10, 11, 12, 13]. They use discrete variables or continuous ones. Among
them, simple contagion models were considered to study the evolution of
distinct social processes, like tax evasion [14], cooperation [15], rumor
spreading [16], epidemic spreading [17], radicalization phenomena [18],
ideological conflicts [19], corruption [20], juvenile crimes [21], obesity
[22].

In this work we consider a fully-connected population composed of $N$
individuals. Agents are labelled in terms of political affiliations, either
left or right. Each group includes three kinds of agents, sensitives,
inflexibles and radicals adding to a total of six subgroups. These subgroups
are denoted sensitive, inflexible, radical. Their respective proportions are
$l$, $ll$, $L$ for the left, and $r$, $rr$, $R$ for the right.

While inflexibles and radicals do not shift their political affiliations,
sensitives are liable to shift to the opposite side via some specific pair
interactions. Accordingly, while proportions $ll$, $L$, $rr$, $R$ are given
and constants, proportions $l$ and $r$ are functions of time. What makes an
individual to become radical or inflexible is out the scope of the present
work. We thus have the conservation condition,

$l(t)+r(t)+L+ll+R+rr=1.$ (1)

We first apply the model to explain the voting outcome for 1989 to 2014
Brazilian elections. After that, we introduce the populists in the model, in
order to give an explanation for the Bolsonaro victory in 2018 election. We
also consider the recent 2021 poll regarding a possible Lula vs Bolsonaro in a
second round in the next year presidential election. We also present an
alternative explanation for the Bolsonaro victory in 2018 election, based on a
single varying parameter without populists. This alternative scenario can
reproduce all election outcomes (1989 - 2018) and the 2021 poll with a single
parameter. For 2018 election and 2021 poll, we compare the results of the two
scenarios, with and without populists. Considering 2018 election, a recent
work by social scientists that discusses the actual impact of populist
attitudes in the election of Bolsonaro is found in agreement with the finding
of our mathematical model [23]. Finally, regarding the two possible scenarios
for the 2018 Bolsonaro victory, we discuss about a hypothetic Lula/Bolsonaro
2022 second round voting and the requirements to yield either Lula or
Bolsonaro victory in 2022.

##  II Competition between left and right over sensitives

During last decades since last World War II, with few exceptions radicals have
been excluded from social discussions about voting choices among people from
left and right. On both sides, left and right, non radicals refused to discuss
with their side-extremists often labelled “fascists” for the right and
“communists” for the left. Discussions were restricted between inflexibles and
sensitives. During these exchanges, sensitives could eventually shift side.
The various interactive configurations with their respective output are,

$\displaystyle l+r$ $\displaystyle\stackrel{{\scriptstyle k}}{{\rightarrow}}$
$\displaystyle l+l,$ (2) $\displaystyle l+r$
$\displaystyle\stackrel{{\scriptstyle 1-k}}{{\rightarrow}}$ $\displaystyle
r+r,$ (3) $\displaystyle ll+r$ $\displaystyle\stackrel{{\scriptstyle
k}}{{\rightarrow}}$ $\displaystyle ll+l,$ (4) $\displaystyle ll+r$
$\displaystyle\stackrel{{\scriptstyle 1-k}}{{\rightarrow}}$ $\displaystyle
ll+r,$ (5) $\displaystyle l+rr$ $\displaystyle\stackrel{{\scriptstyle
1-k}}{{\rightarrow}}$ $\displaystyle r+rr,$ (6) $\displaystyle l+rr$
$\displaystyle\stackrel{{\scriptstyle k}}{{\rightarrow}}$ $\displaystyle l+rr$
(7)

where $0\leq k\leq 1$. For simplicity, we have assumed equal shift
probabilities. For each case, sensitives either shift side or keep on their
current stance. The rate $k$ measures the propensity:

  * •

for a left sensitive to convince a right sensitive and $(1-k)$ the reverse.

  * •

for a left inflexible to convince a right sensitive while $(1-k)$ is the
capacity of a sensitive right to resist this outcome.

  * •

for a left sensitive to resist a right inflexible and $(1-k)$ the reverse.

We also consider equal fixed proportions $f=L=ll=R=rr$ for left and right
inflexibles and radicals. At this stage it is worth stressing that our goal
focuses on getting a minimal coherent frame that yields all the Brazilian
elections’ results to address the question of the impact of populism if any in
Bolsonaro 2018 victory.

The corresponding time evolution of $l(t)$ and $r(t)$ are given by,

$\displaystyle\frac{dl(t)}{dt}$ $\displaystyle=$
$\displaystyle-(1-2k)\,l(t)\,r(t)+k\,f\,r(t)-(1-k)\,f\,l(t),$ (8)
$\displaystyle\frac{dr(t)}{dt}$ $\displaystyle=$
$\displaystyle(1-2k)\,l(t)\,r(t)-k\,f\,r(t)+(1-k)\,f\,l(t),$ (9)

where $\frac{dl(t)}{dt}=-\frac{dr(t)}{dt}$ as expected from the fact that the
dynamics of affiliation occurs only within the group of sensitives with
$l(t)+r(t)=1-ll-L-rr-R=1-4f=constant$.

In such a case, solving the equation $\frac{dl}{dt}=0$ yields the steady
state,

$l=\frac{k\,f\,r}{(1-k)\,f-(2k-1)\,r}$ (10)

where $l=l(t\to\infty)$ and $r=r(t\to\infty)$ are the stationary values of
$l(t)$ and $r(t)$, respectively. From Eq. (1), substituting $r=1-l-4f$ into
Eq. (10) yields a second order polynomial for $l$,

$Al^{2}+Bl+C=0,$ (11)

where

$\displaystyle A$ $\displaystyle\equiv$ $\displaystyle-(1-2k),$ (12)
$\displaystyle B$ $\displaystyle\equiv$
$\displaystyle(1-k)\,f+(1-2k)\,(1-4f)+k\,f,$ (13) $\displaystyle C$
$\displaystyle\equiv$ $\displaystyle-k\,(1-4f)\,f,$ (14)

From Eq. (11) we obtain 2 solutions,

$l=\frac{-[(1-k)\,f+(1-2k)\,4f+k\,f]\pm\sqrt{\Delta}}{2\,(2k-1)}.$ (15)

where $\Delta=B^{2}-4AC$, with $A$, $B$ and $C$ given by Eqs. (12) - (14).
Numerically, we found that the solution with the plus signal is the relevant
one giving $l>0$. The stationary value for the sensitive right fraction $r$
can be obtained from the normalization condition (Eq. (1)), i.e., $r=1-l-4f$.

In particular, the case $k=1/2$ of symmetry balance between shift from
sensitive interactions yields,

$\displaystyle l$ $\displaystyle=$ $\displaystyle\frac{(1-4f)}{2}$ (16)
$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{(1-4f)}{2}.$ (17)

##  III Applying the model to Brazilian elections

After a long dictatorship (1964 - 1985) a democratic system has been installed
based on a two rounds’ presidential elections. Voting is mandatory although
penalty fee for not voting is moderate 111The value vary between $R\$1.05$ and
$R\$3.51$ in Brazilian currency (Reals, where $R\$1$ corresponds to $0.18$
US$\$$).. In 1989, the first president to be elected has been the center-right
candidate Fernando Collor de Mello. Following elections in 1994 and 1998 were
won by the right-wing candidate Fernando Henrique Cardoso at first rounds. The
left-wing candidate Luis Inácio Lula da Silva was elected twice in a row at
second rounds in 2002 and 2006. His successor, Dilma Roussef, was also elected
twice in a row at second rounds in 2010 and 2014. Finally, the far-right
candidate Jair Bolsonaro was elected in 2018 at second round [24].

For the above-mentioned elections, six had a second round (1989, 2002, 2006,
2010, 2014, 2018) and two (1994, 1998) had the president elected at the first
round [24]. Respective voting outcomes are:

  * •

$1989:0.53$ (right-wing candidate) and $0.47$ (left-wing candidate)

  * •

$1994:0.61$ (right-wing candidate) and $0.39$ (left-wing candidate)

  * •

$1998:0.57$ (right-wing candidate) and $0.43$ (left-wing candidate)

  * •

$2002:0.39$ (right-wing candidate) and $0.61$ (left-wing candidate)

  * •

$2006:0.39$ (right-wing candidate) and $0.61$ (left-wing candidate)

  * •

$2010:0.44$ (right-wing candidate) and $0.56$ (left-wing candidate)

  * •

$2014:0.48$ (right-wing candidate) and $0.52$ (left-wing candidate)

  * •

$2018:0.55$ (right-wing candidate) and $0.45$ (left-wing candidate)

Figure 1: (Color online) Time evolution of the densities of sensitives ($l$
and $r$), total right ($r+rr+R$), total left ($l+ll+L$) and the difference
$total\,\,left-total\,\,right$. (a: upper left) Simulation of 1989 election.
(b: upper right) Simulation of 2002 election. (c: lower) Simulation of 2006
election. The black squares indicate the initial conditions, defined by polls
[25]: (a) $l(0)=0.40,r(0)=0.44$; (b) $l(0)=0.56,r(0)=0.28$; (c)
$l(0)=0.45,r(0)=0.39$. We took $f=0.04$ with the value of $k$ for each
election given in Table 1.

Figure 2: (Color online) Time evolution of the densities of sensitives ($l$
and $r$), total right ($r+rr+R$), total left ($l+ll+L$) and the difference
$total\,\,left-total\,\,right$. (a: upper left) Simulation of 2010 election.
(b: upper right) Simulation of 2014 election. (c: lower) Simulation of 2018
election. The black squares indicate the initial conditions, defined by polls
[25]: $l(0)=0.47,r(0)=0.37$; (b) $l(0)=0.44,r(0)=0.40$ and (c)
$l(0)=0.34,r(0)=0.50$. We took $f=0.04$ with the value of $k$ for each
election given in Table 1.

Above results consider only the valid votes, i.e., they do not take into
account null votes or abstentions. Moreover, for the first round 1994
election, the value $0.61$ for the right-wing candidate is obtained summing
outcomes of the actual right-wing winner ($\approx 0.54$) and the third
candidate also right-wing ($\approx 0.07$). Similarly for the first round 1998
election, we added the outcomes of second ($\approx 0.32$ ) and third
($\approx 0.11$) candidates both being left-wing to yield $0.43$ for the left-
wing candidate.

For all those eight elections, fixing $f$ and using Eqs. (15) and (1) we are
able to find appropriate values of $k$ producing steady-state values for total
left$=l+ll+L=l+2f$ and total right$=r+rr+R=r+2f$, which are identical to the
actual voting outcomes.

For the next results, we chose $f=0.04$. For such value, we can recover all
the voting outcomes as a function of the single parameter $k$. The respective
values of $k$ compatible with available data [24] are given in Table 1. We
chose $f=0.04$ since it yields a reasonable value for the total density of
inflexibles and radicals, namely a bit less than $20\%\,(4f=4\times
0.04=0.16)$. Moreover, we found that increasing the total amount to about a
quarter has very little effect on the values of $k$.

$Election$ | $k$
---|---
1989 | 0.497
1994 | 0.488
1998 | 0.493
2002 | 0.513
2006 | 0.513
2010 | 0.507
2014 | 0.502
2018 | 0.495
Table 1: Values of $k$ found to match the elections’ results from 1989 to 2018
considering the model without populists, for fixed $f=L=ll+R=rr=0.04$.

Figures 1 and 2 exhibit the various dynamics obtained from the numerical
integration of Eqs. (8) and (9) using the sets of parameters given in Table 1.
In each figure we plot the densities of sensitives $l$ and $r$, the total
densities of left ($l+ll+L$) and right ($r+rr+R$) supporters, and the
difference $total\,\,left-total\,\,right$. The initial conditions are
represented by the black squares, and they were obtained by polls [25]. The
corresponding values $l(0)$ and $r(0)$ for the initial conditions are given in
the captions of Figures 1 and 2.

The model reproduces both victories of ring-wing and left-wing, depending on
the value of $k$. From the values of $k$, even taking into account that the
radicals do not interact with the other two classes (sensitives and
inflexibles), the presence of such radicals is important.

##  IV The emergence of populism: An alternative explanation for 2018
Bolsonaro victory

###  IV.1 Bolsonaro and Populism

In 2018, the populist Jair Bolsonaro was elected at second round in the
Brazilian presidential election. But why is Bolsonaro considered a populist?
We can adopt the criteria of the Team Populism [26]. The rubric produced by
Team Populism grades the politicians speeches on a scale that goes from $0$ to
$2$: $0$ is classified as “not populist”, $0.5$ as “somewhat populist”, $1.0$
as “populist”, $1.5$ as “very populist” and leaves $2.0$ open for what they
call “perfect populist” [5]. As stated in [5], unlike past presidents Dilma
Rousseff, Lula and Fernando Henrique Cardoso, Bolsonaro articulates stronger
populist elements and uses them more often. His predecessors’ average scores
vary from $0$ to $0.3$, not enough to be considered populist. Based on
Bolsonaro’s speechs, the Team Populism obtained a grade $0.88$ [5].

Thus, we can make a basic and simple hypothesis to include populists among
agents. We assume that populists are former sensitive agents from both left
and right, who had interacted with their respective radicals. Populists form a
seventh subgroup of agents denoted $p(t)$. In addition, once they are created,
populists do interact with sensitives from both left and right. The proportion
of populists is thus a function of time and sensitives. The normalization
condition given by Eq. (1) becomes,

$l(t)+r(t)+p(t)+4f=1.$ (18)

In addition to the social interactions given by Eqs. (2)-(7), we have,

$\displaystyle L+l$
$\displaystyle\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}$ $\displaystyle
L+p,$ (19) $\displaystyle L+l$ $\displaystyle\stackrel{{\scriptstyle
1-\alpha}}{{\rightarrow}}$ $\displaystyle L+l,$ (20) $\displaystyle R+r$
$\displaystyle\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}$ $\displaystyle
R+p,$ (21) $\displaystyle R+r$ $\displaystyle\stackrel{{\scriptstyle
1-\alpha}}{{\rightarrow}}$ $\displaystyle R+r,$ (22) $\displaystyle l+p$
$\displaystyle\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}$ $\displaystyle
p+p,$ (23) $\displaystyle l+p$ $\displaystyle\stackrel{{\scriptstyle
1-\alpha}}{{\rightarrow}}$ $\displaystyle l+l,$ (24) $\displaystyle r+p$
$\displaystyle\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}$ $\displaystyle
p+p,$ (25) $\displaystyle r+p$ $\displaystyle\stackrel{{\scriptstyle
1-\alpha}}{{\rightarrow}}$ $\displaystyle r+r,$ (26)

Keeping on with our simplified approach we chose the same parameter $\alpha$
for the various probabilistic shifts. We now have two varying parameters,
namely $k$ and $\alpha$. Above interactions rules lead to the evolution
equations,

$\displaystyle\frac{dl(t)}{dt}$ $\displaystyle=$
$\displaystyle-(1-2k)\,l(t)\,r(t)+k\,f\,r(t)-(1-k)\,f\,l(t)-\alpha\,f\,l(t)$
(27) $\displaystyle+(1-2\alpha)\,l(t)\,p(t),$ $\displaystyle\frac{dr(t)}{dt}$
$\displaystyle=$
$\displaystyle(1-2k)\,l(t)\,r(t)-k\,f\,r(t)+(1-k)\,f\,l(t)-\alpha\,f\,r(t)$
(28) $\displaystyle+(1-2\alpha)\,r(t)\,p(t),$ $\displaystyle\frac{dp(t)}{dt}$
$\displaystyle=$
$\displaystyle\alpha\,f\,l(t)+\alpha\,f\,r(t)-(1-2\alpha)\,l(t)\,p(t)-(1-2\alpha)\,r(t)\,p(t).$
(29)

Figure 3: (Color online) Time evolution of the densities. These results
simulate the evolution of votes in the 2018 election considering the populism
coupling. We observe the Bolsonaro victory considering the total right
($r+rr+R$) plus the populists $p$, i.e., for $r+rr+R+p$. (a, left panel:)
results $\alpha=0.30$. (b, right panel:) results for $\alpha=0.40$. The black
squares in both panels denote the initial conditions, obtained by polls [25],
i.e., the same values considering in Figure 2 (c), $l(0)=0.34$ and
$r(0)=0.50$. We have $f=0.04$ with and the value of $k$ for each election
given in Table 2.

We performed a numerical integration of the evolution equations (27) - (29) in
order to obtain the 2018 election results, namely Bolsonaro received about
$55\%$ of the votes and the left-wing candidate Haddad received about $45\%$
of the votes [24]. As in the previous section $f=0.04$. In Figure 3 we exhibit
two graphics showing the time evolution of the densities of interest, for the
choices $\alpha=0.30$ (left panel) and $\alpha=0.40$ (right panel). For both
graphics, we consider the same initial condition given by polls [25], namely
$l(0)=0.34$ and $r(0)=0.50$ (the same considered in section III). In addition,
we also consider $p(0)=0$. To match the 2018 election’s results, we found
$k=0.496$ for $\alpha=0.30$ and $k=0.499$ for $\alpha=0.40$. In such a case,
based on the above discussion about the Bolsonaro’s populism, we considered
the votes from the populist compartiment to Bolsonaro, i.e., the total
fraction of votes for the right candidate is given by $r+rr+R+p$.

Some comments about the results:

  * •

Comparing with the 2014 election, the left-wing candidate support decreased
from $0.52$ in 2014 to $0.45$ in 2018. In addition, the right-wing candidate
support increased from $0.48$ in 2014 to $0.55$ in 2018.

  * •

Based on the model, considering as a case study the value $\alpha=0.40$, we
obtained the equilibrium values total left $=0.45$, total right $=0.47$ and
$p=0.08$, leading to total right plus populists $=0.55$. In other words, the
fraction of right individuals decreased from $0.48$ in 2014 to $0.47$ in 2018.

  * •

However, the fraction of populists increased from zero in 2014 to $0.08$ in
2018. Those choices are based on the assumption that right sensitives $r$
turned in populists $p$ from interacting with few right radicals $R$, and once
a seed of populists $p$ is formed it sucks directly a part of right sensitives
$r$ and also a small part of left sensitives $l$.

###  IV.2 Comment regarding the 2021 poll about a possible Bolsonaro vs Lula
in 2022

A recent poll was conducted in Brazil in September 2021 about a possible 2022
second round Bolsonaro / Lula (right / left). The results yielded a
substantial advantage for Lula with $0.64$ against $0.36$ for Bolsonaro [27],
a large shift of $0.19$ with respect to the 2018 results.

Within our model, such a shift implies (1) A decrease of the populism support,
that was $p=0.08$ for 2018 (for $\alpha=0.40$); (2) A large number of right
sensitives $r$ turning into left sensitives $l$. Indeed, an increase in $k$
with a simultaneous decrease of $\alpha$ can produce the poll observed shift
of $0.19$ from right to left.

##  V Comparing the two explanations for 2018 Bolsonaro victory

To explain 2018 Bolsonaro victory, we have first considered the usual
competition between left and right for sensitives, without interactions with
radicals (model of section II). After that, in section IV.A we introduced the
mechanism of interaction between sensitives and radicals to create populists,
who then can spread over interacting with sensitives from both left and right.
Then, to recover Bolsonaro outcome of $0.55$ we found the equilibrium values
$p=0.03$ (for $\alpha=0.30$) and $p=0.08$ (for $\alpha=0.40$) for populists,
which, contrary to what was expected for a populist leader, is not that high.

$Election$ | $k$
---|---
2018 | 0.495
2018 | 0.4961
2018 | 0.4992
2021 | 0.516
2021 | 0.5201
2021 | 0.5282
Table 2: Values of $k$ found to reproduce the elections’ results considering
the model without populists (model of Section II) and the model with populists
(model of Section IV.A) for $\alpha=0.30$ (values with 1) and $\alpha=0.40$
(values with 2).

Figure 4: (Color online) (a: upper left) Values of $k$ matching the elections
results with no populism. The values with populism are added for 2018 and
2021. The x-axis is the time (in years) since the first election: $0$
represents the 1989 election, $5$ represents the 1994 election, and so on. (b:
upper right, c: lower left, d: lower right) The plots show the stationary
densities of populists $p$, left sensitives $l$ and right sensitives $r$ as a
functions of $\alpha$ with $f=0.04$. The symbols are numerical results, and
the lines are just guides to the eye. In panel (b), the inset exhibits the
same data for $p$ only with symbols (circles, squares and triangles), together
with the analytical result of Eq. (30) (full black line).

In Table 2 we exhibit the values of $k$ considered to match the 2018 election
as well as the 2021 poll. For comparison, we included in Tab. 2 the values of
$k$ obtained for the model without populists. For $\alpha=0.30$ we get
respectively $k=0.496$ and $k=0.520$ (see Table 2) and the same equilibrium
value of populists $p=0.03$. When $\alpha=0.40$ we found $k=0.499$ and
$k=0.528$ and still a unique $p=0.08$ (see Table 2). All the results are
exhibited Figure 4 (a, b).

These small values of the proportions of populists are unexpected and puzzling
when dealing with a candidate pictured as populist. To check this “anomaly” we
look at what happens increasing the populist coupling to $\alpha=0.50$. The
increase of only $0.10$ reveals a sudden sharp jump in the equilibrium value
of populists at $p=0.84$ against $p=0.08$ for $\alpha=0.40$.

Above jump in the value of populists hints at a singularity at some value of
the populist coupling. To identify this singularity we plot in Figure 4 (b, c,
d) the variations of stationary fractions of populists $p$, left sensitives
$l$ and right sensitives $r$ as a function of $\alpha$ for three different
values $k=0.10,0.40,0.70$. Indeed, abrupt quasi-vertical jump and falls are
seen to occur at $\alpha=0.50$.

In particular, Figure 4 (b) shows that the increase of $p$ is very slow and
stays at low values till the vicinity of $\alpha=0.45$, before a sudden jump
to each high values. This behavior sheds a new light on the phenomenon of
populism, which could explain why it is almost impossible to oppose it when it
bursts. Indeed, since the impact of populism is minor on the opinion dynamics
during about half the range of the associated parameters, it generates a low
proportional reaction, which in turn is overwhelmed by the tidal wave populism
when it reaches its breaking point.

These numerical results regarding the stationary fraction of populists only as
a function of $\alpha$ can be understood if we inspect Eq. (29). In the
stationary state we have $dp/dt=0$, and Eq. (29) leads to the expression
$[(1-2\alpha)\,(l+r)]p=\alpha\,f\,(l+r)$. Thus, we obtain

$p=p(\alpha)=f\,\frac{\alpha}{1-2\alpha}~{}.$ (30)

This result is valid for $\alpha<0.50$, $l\neq 0$ and $r\neq 0$. Eq. (30)
shows that the stationary values of $p$ does not depend on $k$, as we
discussed before. Indeed, $p$ is solely a function of $\alpha$, and the
function $p(\alpha)$ has a jump at $\alpha=0.50$. Eq. (30) explains our
previous observations, and it is exhibited in the inset of Figure 4 (b)
together with the numerical results for the stationary $p$ obtained from the
numerical integration of Eqs. (27) - (29). In addition, as we observed in
Figure 4 (c,d) that $l=r=0$ for $\alpha>0.50$, from the normalization
condition (18) we have $p=1-4f$. For the case of Figure 4 (b), we have
$f=0.04$ and thus $p=0.84$ for $\alpha>0.50$.

Above results lead to question the actual narrative which explains Bolsonaro
2018 victory as a populist event. At this stage, our analysis yields two
possible scenarios with very different underlying political mechanisms:

  1. 1.

Bolsonaro 2018 victory was not the result of populism but the outcome of the
regular competition between sensitives from left and right.

  2. 2.

Bolsonaro did benefit from populism but at a very low level with only a few
percent.

Such discussion can be connected with a recent work discussing whether
Bolsonaro’s victory is a remarkable case or a new effective model of public
policy and political leadership for Brazil and other countries in the region
[28]. The author concluded that it will depend largely on Bolsonaro’s success
as the President of Brazil: at the moment, the interim results of his
presidency are highly controversial and rather continue to divide and polarize
Brazilian society.

The results of our mathematical model agree with a recent analysis performed
by social scientists. Based on results from surveys, the authors concluded
that support for the far-right candidate Jair Bolsonaro is explained by right-
wing ideology and illiberal attitudes, with populist attitudes playing a small
role, if any [23].

##  VI Anticipating the 2022 election

In case above second scenario is valid, to anticipate the 2022 outcome we need
to stress that as long as $\alpha<0.50$, the equilibrium value of $p$ stays
low and what matters is the value of $k$. However, as soon as $\alpha\geq
0.50$ the value of $k$ becomes irrelevant, and in such a case the populism
would contribute strongly to Bolsonaro making likely his victory in 2022.

That would make the 2021 poll misleading to forecast the 2022 outcome since it
was implemented with low populism couplings putting Bolsonaro in a status of
an hidden winner since during the 2022 campaign these couplings could be
increased moderately to reach the jump of $p$ as seen from Figure 4 (b) and
then propels Bolsonaro to victory. That would result in a surprising reversing
of the 2021 Lula tilde like poll outcome in 2022 at the benefit of Bolsonaro.

##  VII Final Remarks

In this work, using a contagious-like model we have studied the dynamics of
opinion to describe the evolution of the political voting trends in Brazil
after the dictatorship from 1985 to nowadays. The model articulates around a
system of coupled ordinary differential equations to account for the
interactions of left and right supporters. For each side we have considered
three kinds of agents, sensitives, inflexibles and radicals where only
sensitives can shift opinions.

The central question we addressed in our manuscript was wether it is necessary
to introduce a new type of populist political orientation to yield 2018
Bolsonaro victory as claimed by many analyses after his election [23, 29]. Our
results have shown that this condition is not necessary to yield Bolsonaro
victory. Moreover, we found that if despite this finding we do introduce a new
populist component, its impact on the 2018 outcome was marginal. However in
this case, this marginal component could render possible a second Bolsonaro
victory in 2022.

Such conclusions were obtained using two distinct settings of the model.
First, we have assumed that radicals do not interact. They are kept apart from
sensitives and inflexibles. The evolution of vote intentions are then ruled by
interactions among sensitives and inflexibles, as well as among sensitives
themselves. The model was shown to reproduce all elections outcomes after
Brazilian dictatorship (1989 - 2018) using one single parameter $k$ with some
fixed densities of inflexibles and radicals, that we considered equal for
simplicity.

In a second step we have activated interactions of radicals with sensitives to
create populists from sensitives. Once created, populists interact with
sensitives to accelerate their spreading.

Regarding the 2018 election, along the first approach driven solely by $k$, we
showed that we can recover the 2018 outcome as well as the 2021 poll. Our
hypothesis would mean that despite Bolsonaro populist narrative [5], his
victory was the result of the same mechanisms, which have been at play during
all past presidential elections.

In our study, we found two scenarios of very different political contents,
which could have equally led to Bolsonaro victory. While this finding has no
effect on the the past 2018 election, it could put at stake the current belief
that Bolsonaro will lose the 2022 election in case the second scenario is
valid. Moreover, if it is the first scenario, according to our findings, the
2021 poll has overestimated Lula outcome.

Indeed, the two possible scenarios for 2018, with and without populists, were
discussed in a recent paper by social scientists [23]. They tested some
hypothesis about the effects of populist attitudes and ideology on vote choice
in 2018 election. The results suggest that the support for the far right
populist candidate Bolsonaro is explained by right-wing ideology and illiberal
attitudes, with populist attitudes playing a small role, if any. Their
conclusions are in agreement with our findings that populism may not be the
reason of past Bolsonaro victory and if it was, it contributed only at the
margin. On this basis, we argue that a significant source of populism is
potentially available for Bolsonaro at next election.

## Acknowledgments

NC acknowledges financial support from the Brazilian scientific funding
agencies CNPq (Grant 310893/2020-8) and FAPERJ (Grant 203.217/2017). We also
thank L. Sigaud for a critical reading of the manuscript.

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# Modules over some group rings, having $d$-generator property

V.A. Bovdi  UAEU, Al-Ain, United Arab Emirates<EMAIL_ADDRESS>and  L.A.
Kurdachenko  Department of Algebra and Geometry, School of Mathematics and
Mechanics, National University of Dnipro, Ukraine<EMAIL_ADDRESS>
###### Abstract.

For modules over group rings we introduce the following numerical parameter.
We say that a module $A$ over a ring $R$ has finite $r$-generator property if
each f.g. (finitely generated) $R$-submodule of $A$ can be generated exactly
by $r$ elements and there exists a f.g. $R$-submodule $D$ of $A$, which has a
minimal generating subset, consisting exactly of $r$ elements. Let $FG$ be the
group algebra of a finite group $G$ over a field $F$. In the present paper
modules over the algebra $FG$ having finite generator property are described.

###### Key words and phrases:

module over a group ring, $r$-generator property, width of a module

###### 1991 Mathematics Subject Classification:

20C05; 20F50 Secondary: 20D25; 16D10; 16D70; 16D80

The research was supported by the UAEU UPAR Grant G00002160

##  1\. Introduction and main results

Let $R$ be an associate ring and let $A$ be a module over the ring $R$. The
structure and the properties of $A$ essentially depend on the properties of
the family of its finitely generated submodules. Clearly, if $B$ is a finitely
generated submodule of $A$ then $B$ can have different generator systems.
Therefore, it makes sense to consider a minimal system of generators. Recall
that a subset $M\subseteq B$ is called a minimal generating subset for $B$ if
$M$ generates $B$ and each proper subset of $M$ generates a proper submodule
of $B$. If $R$ is a field and $A$ is a vector space, then the number of
elements in each minimal system of generators for $B$ is the same, i.e. it is
an invariant of $B$. Generally such a situation does not always take place.
Moreover, there are not so many rings for which this holds. Therefore, in the
submodule $B$, we consider those minimal systems of generators that have the
least number of elements. The number of elements in such minimal system of
generators is denoted by $d_{R}(A)$.

If $A$ is a vector space over a field $F$, then $d_{F}(A)$ is exactly the
dimension of $A$. However, when moving from vector spaces to modules, the
properties of the numerical invariant $d_{R}(A)$ can already differ
significantly from the classical properties of the dimensions of vector
subspaces. For example, if $B$ is a proper subspace of the the vector space
$A$, then $\dim_{F}(B)<\dim_{F}(A)$. Unlike vector spaces, not each submodule
of a finitely generated module is finitely generated. Moreover, in the case
when a submodule is finitely generated it can has a minimal generating set
with more elements than the number of elements in the generating system of the
entire module. The following simple example illustrates this situation.

Let $p$ be a prime. Let $G=\langle g\rangle\times\langle h\rangle\cong
C_{p}\times C_{p}$ be a multiplicative $p$-group of order $p^{2}$, $A=\langle
b\rangle\oplus\langle c\rangle\oplus\langle
d\rangle\cong\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ an
additive elementary abelian $p$-group of order $p^{3}$ and let $H=\langle
c\rangle\oplus\langle d\rangle\leq A$. Define a right action $\circ$ of $g$
and $h$ on $A$ (concerning the basis $\\{b,c,d\\}$) by the following rule:

$a\circ g=\begin{pmatrix}b\\\ c\\\ d\\\ \end{pmatrix}\begin{pmatrix}1&0&0\\\
1&1&0\\\ 0&0&1\\\ \end{pmatrix}\quad\text{and}\quad a\circ
h=\begin{pmatrix}b\\\ c\\\ d\\\ \end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
1&0&1\\\ \end{pmatrix},\qquad(a\in A).$

In other words $b\circ g=b+c$, $c\circ g=c$, $d\circ g=d$, $b\circ h=b+d$,
$c\circ h=c$ and $d\circ h=d$. Clearly, $A=\langle b\rangle_{\mathbb{Z}_{p}G}$
is a cyclic right $\mathbb{Z}_{p}G$-module, in which
$d_{\mathbb{Z}_{p}G}(A)=1$, but $d_{\mathbb{Z}_{p}G}(H)=2$.

Each ring $R$ and their ideals are itself the modules over $R$. Therefore, it
is natural to consider first the number of generating elements (left or right)
ideals in the ring $R$. It is clear that it makes sense to do this for the
rings whose (left or right) ideals are finitely generated, i.e. for the
Noetherian rings. In [7] those commutative rings $R$ were considered, for
which there exists a positive integer $r$ such that $d_{R}(I)\leq r$ for each
ideal $I$ of $R$. It is immediately clear that all examples of such rings are
principal ideal domains and Dedekind domains. Commutative rings with such
property actively studied in [7, 14, 15, 16, 18, 21, 22, 23]. Gilmer [14]
generalized this situation, where he considered commutative rings for which
there exists a positive integer $r$ such that $d_{R}(I)\leq r$ for every
finitely generated ideal $I$ of $R$. Such rings are called rings with $r$ \-
generator property. The most famous examples of such rings (see, for example,
[13, Chapter II.1, Chapter III.5]) are valuation domains and Bezout domains
(i.e. domains in which every finitely generated ideal is principal).
Commutative rings having $r$-generator property have been studied in [14, 15,
16, 18, 21, 22, 23]. In group theory, an analogue of this concept arose
earlier in the work of Maltsev [20]. Following Maltsev, we say that a group
$G$ has finite special rank $r$, where $r$ is a positive integer, if $d(H)\leq
r$ for every finitely generated subgroup $H$ of the group $G$. In group theory
this concept turned out to be very useful and productive (for example, see the
survey [10] and the book [11]). Thus, it is a native to consider an analog of
this natural concept for modules as well. Let $A$ be a module over a ring $R$.
We say that $A$ has $d$-$R$-generator property if there exists a positive
integer $d$ such that $d_{R}(B)\leq d$ for each finitely generated
$R$-submodule $B$ of $A$. If we consider only one fixed ring $R$, then we
simply say that $A$ has $d$-generator property.

In [5] modules were studied which have $d$-generator property over the group
algebra $FG$, where $F$ is a field of characteristic 0 and $G$ is a finite
group. In the paper mentioned above we used another term - ”special rank”, but
in ring theory there are already several terms using the word rank, so in this
paper we preferred to use the term introduced by R. Gilmer for the rings.

In the present paper we study modules having $d$-generator property over the
group algebras $FG$ where $F$ is a field of arbitrary characteristic and $G$
is a finite group. The main results are the following.

######  Theorem 1.

Let $r\in\mathbb{N}$ and let $FG$ be the group algebra of a finite group $G$
over a field $F$. An $FG$-module $A$ has a finite $r$-generator property if
and only if the following conditions hold:

  * (i)

$\dim_{F}(A)$ is finite;

  * (ii)

each semisimple homogeneous $FG$-factor of $A$ has width at most $r$ and there
exists a semisimple homogeneous $FG$-factor, whose width is exactly $r$.

######  Theorem 2.

Let $n,r\in\mathbb{N}$ and let $FG$ be the semisimple group algebra of a
finite group $G$ over a field $F$ of prime characteristic $p$. An $FG$-module
$A$ has finite $r$-generator property if and only if the following conditions
hold:

  * (i)

$A=\oplus_{j=1}^{n}H_{j}$ in which each $H_{j}=\oplus_{k=1}^{t(j)}D_{k}$,
where each $D_{k}$ is a simple $FG$-submodule of $A$, $|G|\leq t(j)$ and
$D_{k}\cong_{FG}D_{m}$ for all $1\leq k,m\leq t(j)$;

  * (ii)

$t(j)\leq r$ for all $j$ and there exists $s\in\mathbb{N}$ such that $t(s)=r$;

  * (iii)

$n\leq nns_{F}(G)$, where $nns_{F}(G)$ is the number of pairwise non-
isomorphic simple $FG$-modules.

######  Theorem 3.

Let $m,r\in\mathbb{N}$ and let $FG$ be the modular group algebra of a finite
group $G$ over a field $F$ of prime characteristic $p$. Let $P\in Syl_{p}(G)$
be a normal Sylow $p$-subgroup of $G$. If an $FG$-module $A$ has finite
$r$-generator property, then the following conditions hold:

  * (i)

$\dim_{F}(A)$ is finite;

  * (ii)

$A$ has a series of $FG$-submodules

$\langle 0\rangle=Z_{0}\leq Z_{1}\leq\cdots\leq Z_{m-1}\leq Z_{m}=A$

such that $m\leq|P|$ and each $G/C_{G}(Z_{j+1}/Z_{j})$ is a $p$-group, each
factor $Z_{j+1}/Z_{j}$ is a semisimple $FG$-module, whose homogeneous
components have width at most $r$ and the number of homogeneous components is
at most $nns_{F}(G/P)$, where $nns_{F}(G)$ is the number of pairwise non-
isomorphic simple $FG$-modules.

##  2\. Preliminaries and Lemmas

In the sequel, we use the notions and results from the books [4, 9, 11].

######  Lemma 2.1.

Let $A$ be an $R$-module over a ring $R$ such that $A$ has finite
$r$-generator property. If $B,C$ are $R$-submodules of $A$ such that
$B\leq_{R}C$ then the following conditions hold:

  * (i)

the submodule $B$ has finite $r$-generator property at most $r$;

  * (ii)

the factor-module $A/B$ has finite $r$-generator property at most $r$;

  * (iii)

the factor-module $C/B$ has finite $r$-generator property at most $r$.

These assertions are obvious.

######  Lemma 2.2.

Let $r,t\in\mathbb{N}$ and let $B$ be a submodule of an $R$-module $A$ over a
Noetherian ring $R$. If $B$ has finite $r$-generator property at most $r$ and
$A/B$ has finite $d$-generator property at most $t$, then $A$ has
$d$-generator property at most $r+t$.

###### Proof.

Let $D$ be a f.g. $R$-submodule of $A$. Clearly, $D$ is Noetherian as an f.g.
module (see [19, Lemma 1.1]). The factor $D/(D\cap B)\cong(D+B)/B$ has finite
generating subset $\\{d_{j}+(D\cap B)\mid 1\leq j\leq m\\}$ in which $m\leq
t$. It follows that $D\cap B$ is a f.g. $R$-submodule of $B$. Since $B$ has
$r$-generator property, $D\cap B$ has finite generating subset $\\{b_{j}\mid
1\leq j\leq k\\}$ where $k\leq r$. The subset $\\{d_{j}\mid 1\leq j\leq
m\\}\cup\\{b_{j}\mid 1\leq j\leq k\\}$ generates $D$ as an $R$-submodule and
$m+k\leq t+r$. ∎

######  Lemma 2.3.

Let $A=\oplus_{j=1}^{n}A_{j}$ be an $R$-module over a ring $R$ in which each
$A_{j}$ is a simple $R$-submodule. If elements of the set
$\\{Ann_{R}^{left}(A_{1}),\ldots,Ann_{R}^{left}(A_{n})\\}$ are pairwise non-
isomorphic, then $A$ is a cyclic $R$-module.

###### Proof.

Let us prove that $A=\langle a_{1}+\cdots+a_{n}\mid 0\not=a_{j}\in
A_{j}\rangle_{R}$. Since each $A_{j}$ is a simple $R$-module,
$U_{j}:=Ann_{R}^{left}(A_{j})$ is a maximal left ideal of $R$. The fact that a
sum of left ideals is also a left ideal together with $U_{j}\not=U_{m}$
implies that $U_{j}+U_{m}=R$ for all $j\not=m$. Thus $U_{1}+U_{2}=R$ and there
exists $\alpha\in U_{2}\setminus U_{1}$ such that $\alpha a_{1}\not=0$,
$\alpha a_{2}=0$ and

$\alpha(a_{1}+\cdots+a_{n})=\alpha a_{1}+(\alpha a_{3}+\cdots+\alpha a_{n}).$

If $\alpha a_{3}+\cdots+\alpha a_{n}=0$, then $\alpha a_{1}\in
R(a_{1}+\cdots+a_{n})$. Since $A_{1}$ is a simple $R$-module, it can be
generated by each of its non-zero elements, so $A_{1}=\langle
a_{1}\rangle_{R}\leq\langle a_{1}+\cdots+a_{n}\rangle_{R}$.

Let $\alpha a_{3}+\cdots+\alpha a_{n}\not=0$ and let $j_{1}\in\mathbb{N}$ be a
smallest integer (say $j_{1}=3$) such that $j_{1}>2$ and $\alpha
A_{j_{1}}\not=0$. The equality $U_{1}+U_{3}=R$ shows that there exists
$\beta\in U_{3}\setminus U_{1}$ such that $\beta\alpha a_{1}\not=0$,
$\beta\alpha a_{3}=0$ and

$\beta(\alpha a_{1}+\alpha a_{3}+\cdots+\alpha an)=\beta\alpha
a_{1}+(\beta\alpha a_{4}+\cdots+\beta\alpha a_{n}).$

If $\beta\alpha a_{4}+\cdots+\beta\alpha a_{n}=0$, then, as above,
$A_{1}\leq\langle a_{1}+\cdots+a_{n}\rangle_{R}$. Otherwise, all this process
we can repeat again. Since $n\in\mathbb{N}$, after finitely many steps we
obtain that $A_{1}\leq R(a_{1}+\cdots+a_{n})$ and $A=A_{1}\oplus\cdots\oplus
A_{n}=\langle a_{1}+\cdots+a_{n}\rangle_{R}$. ∎

######  Lemma 2.4.

Let $A=\oplus_{j=1}^{n}A_{j}$ be an $R$-module over a ring $R$ in which each
$A_{j}$ is a simple $R$-submodule. If all elements of the set
$\\{Ann_{R}^{left}(A_{1}),\ldots,Ann_{R}^{left}(A_{n})\\}$ coincide, then
$Ann_{R}^{left}(A_{j})=Ann_{R}^{left}(\langle a\rangle_{R})$ for each $a\in
A\setminus\\{0\\}$. In particular, the $R$ -submodule $\langle a\rangle$ is
isomorphic to each $A_{j}$ for $1\leq j\leq n$.

###### Proof.

Set $L:=Ann_{R}^{left}(A_{1})$. Clearly, $L\leq Ann_{R}^{left}(a)$ for each
$0\not=a\in A$, so that $L\leq Ann_{R}^{left}(A)$. Since $A_{1}$ is a simple
$R$-module, $L$ is a maximal left ideal of $R$. It follows that
$L=Ann_{R}^{left}(A)$.

Let $y\in R\setminus L$ and let $a=a_{1}+\cdots+a_{n}\in A\setminus\\{0\\}$,
in which each $a_{j}\in A_{j}$, $a_{1}\not=0$ and $Ann_{R}^{left}(a)=R$. Since
$ya=0$ and $a_{1},\ldots,a_{n}$ are linearly independent over $R$, $yA_{j}=0$
for each $j$. In particular, $ya_{1}=0$ and $y\in Ann_{R}^{left}(A_{1})=L$, a
contradiction. It follows that $Ann_{R}^{left}(a)\not=R$ for each $a\in
R\setminus\\{0\\}$. Consequently, $Ann_{R}^{left}(a)=L$ and $\langle
b\rangle_{R}\cong_{R}R/L$ for each $b\in R\setminus\\{0\\}$. ∎

######  Corollary 2.5.

Let $\Lambda$ be a set and let $A=\oplus_{\lambda\in\Lambda}A_{\lambda}$ be an
$R$-module over a ring $R$ in which each $A_{\lambda}$ is a simple
$R$-submodule. If

$Ann_{R}^{left}(A_{\lambda})=Ann_{R}^{left}(A\mu),\qquad(\lambda,\mu\in\Lambda)$

then each cyclic $R$-submodule of $A$ is simple and
$A_{\lambda}\cong_{R}\langle a\rangle_{R}$ for each $a\in A\setminus\\{0\\}$.

######  Corollary 2.6.

Let $n,r\in\mathbb{N}$ and let $A=\oplus_{j=1}^{n}A_{j}$ be an $R$-module over
a ring $R$ in which each $A_{j}$ is a simple $R$-submodule.

If all elements of the set
$\\{Ann_{R}^{left}(A_{1}),\ldots,Ann_{R}^{left}(A_{n})\\}$ coincide, then the
following conditions are equivalent:

  * (i)

$d_{R}(A)=r$;

  * (ii)

$A$ has finite $r$-generator property;

  * (iii)

$n=r$, where $n={\rm width}_{R}(A)$.

###### Proof.

$(i)\Leftrightarrow(iii)$. Let $S:=\\{b_{1},\ldots,b_{k}\\}\subseteq A$ be a
minimal set, such that $A=\langle S\rangle_{R}$ and let $B_{j}:=\langle
b_{j}\rangle_{R}$. Obviously, $B_{j}\cong_{R}A_{j}$ after a suitable
renumbering of the set $\\{A_{k}\\}$ and $B_{j}$ is simple for each $j$ by
Lemma 2.4. Since $A$ is a semisimple $R$-module, $A$ includes an $R$-submodule
$D_{1}$ such that $A=B_{1}\oplus D_{1}$ (see [19, Corollary 4.3]). If
$b_{t}\in B_{1}$ for some $t>1$, then $A=\langle S\setminus b_{t}\rangle_{R}$,
a contradiction. This yields that $b_{t}\not\in B_{1}$. Similarly, if $t>2$
then $b_{t}\not\in B_{1}\oplus B_{2}$. Repeating this process finitely many
times, we obtain $b_{k}\not\in B_{1}\oplus\cdots\oplus B_{k-1}$ and
$A=\oplus_{j=1}^{k}B_{j}$.

If $T:=\\{d_{1},\ldots,d_{m}\\}\subseteq A$, such that $S\not=T$ and
$A=\langle T\rangle_{R}$, then repeating the above arguments, we obtain a
decomposition $A=\oplus_{j=1}^{m}D_{j}$ in which each $D_{j}=\langle
d_{j}\rangle_{R}\cong_{R}A_{j}$ after a suitable renumbering of the set
$\\{A_{k}\\}$. It follows that $k=m$ by the Krull-Remak-Schmidt theorem (see
[8, Chapter 6, Theorem 1.6]) and $d_{R}(A)=n$.

$(ii)\Leftrightarrow(iii)$. Since $A$ is a f.g. $R$-module, $d_{R}(A)=r$. Thus
$n=r$ by the part above.

Conversely, let $n=r$. Using part (i) we obtain that $d_{R}(A)=r$. Let B be a
f.g. $R$-submodule of A. Since $A$ is a semisimple $R$-module, there exists an
$R$-submodule C such that $A=B\oplus C$ (see [19, Corollary 4.3]). Since $B,C$
are semisimple, $B=\oplus_{j=1}^{s}B_{j}$ and $C=\oplus_{j=1}^{t}C_{j}$, where
each $B_{j}$ and $C_{j}$ are simple $R$-submodules (see [19, Corollary 4.3]).
Consequently, using the Krull-Remak-Schmidt theorem, we get

$A=(\oplus_{j=1}^{s}B_{j})\oplus(\oplus_{j=1}^{t}C_{j}),$

in which $s+t=r$ and $B_{j}\cong_{R}A_{1}$, so $d_{R}(B)=s\leq r$ by part (i).
∎

Let $A=\oplus_{j=1}^{n}A_{j}$ be an $R$-module over a ring $R$ in which each
$A_{j}$ is a simple $R$-module. The relation ”$A_{j}\cong_{R}A_{m}$” is an
equivalence relation on the set $\\{A_{j}\mid 1\leq j\leq n\\}$. Let
$E_{1},\ldots,E_{k}$ be all equivalence classes by its relation. A direct sum
of all submodules from the set $E_{j}$ we denote by $S_{j}$ and it is called a
homogeneous component of A.

######  Corollary 2.7.

Let $n,r\in\mathbb{N}$ and let $A=\oplus_{j=1}^{n}A_{j}$ be a semisimple
$R$-module over a ring $R$ in which each $A_{j}$ is a simple $R$-module. The
following conditions are equivalent:

  * (i)

$d_{R}(A)=r$;

  * (ii)

$A$ has $r$-generator property;

  * (iii)

all homogeneous components of $A$ have width at most $r$ and there exists a
homogeneous component, having width exactly $r$.

###### Proof.

$(i)\Leftrightarrow(iii)$. Let $d_{R}(A)=r$. Let $\\{E_{1},\ldots,E_{k}\\}$ be
a collection of all homogeneous components of $A$. Set
$D_{m}=\oplus_{j=1,j\not=m}^{n}E_{j}$. Clearly, $A=\oplus_{j=1}^{k}E_{j}$ and
$A/D_{m}\cong_{K}E_{m}$ is a semisimple homogeneous module, so

$d_{R}(A/D_{m})\leq d_{R}(A)=r,\qquad(1\leq m\leq n)$

by Lemma 2.1(ii). Since $d_{R}(E_{m})={\rm width}_{R}(E_{m})$ by Corollary
2.6(i)(iii),

${\rm width}_{R}(E_{m})\leq r.$

Without lost of generality we can assume that $E_{1}$ has a greatest width,
say
${\rm width}_{R}(E_{1})=s$ with $s\in\mathbb{N}$. Furthermore,
$E_{1}=\oplus_{j=1}^{s}C_{j}$ in which each $C_{j}$ is a simple $R$-submodule
whose homogeneous components have width $1$ by Corollary 2.6(iii). Each module
$C_{j}$ is cyclic by Lemma 2.4, so $A$ has a set of generators consisting of
$s$ elements. Hence $s\leq r$. The above proved inequality $r\leq s$ implies
that $r=s$. Using similar arguments, the conditions of our lemma are
sufficient.

$(ii)\Leftrightarrow(iii)$. Since $A$ is a f.g. $R$-module, $d_{R}(A)=r$, so
we can repeat the proof of part $(i)\Rightarrow(iii)$.

Conversely, $(iii)\Rightarrow(i)$ as was proved before, so $d_{R}(A)=r$. Let
$B\leq_{R}A$ be a f.g. $R$-submodule. Since $A$ is a semisimple $R$-module,
there exists an $R$-submodule $C$ such that $A=B\oplus C$ (see [19, Corollary
4.3]) with semisimple components $B$ and $C$ (see [19, Corollary 4.3]). Thus

$A=B\oplus C=(\oplus_{j=1}^{s}B_{j})\oplus(\oplus_{j=1}^{t}C_{j}),$

where each $B_{j}$ and $C_{j}$ are simple $R$-submodules. Finally, according
to the Krull-Remak-Schmidt theorem (see [8, Chapter 6, Theorem 1.6]), each
homogeneous component of $B$ has width at most $r$, so $d_{R}(B)\leq r$ by
part (iii).∎

######  Lemma 2.8.

Let $A$ be an $FG$-module, where $FG$ is the group algebra of a finite group
$G$ over a field $F$. If $A$ has finite $r$-generator property, then
$\dim_{F}(A)\leq r|G|$.

###### Proof.

Let $0\not=a\in A$. Clearly, $\langle
a\rangle_{FG}\cong_{F}FG/Ann_{FG}^{left}(a)$ and

$\dim_{FG}(\langle a\rangle_{FG})=\dim_{F}((FG)a)\leq\dim_{F}(FG)=|G|.$

Let $B_{1}\leq_{FG}A$ such that $B_{1}$ is f.g. and $B_{1}\not=A$. Since $A$
has $r$-generator property, there exists a finite subset
$M=\\{a_{1},\ldots,a_{m}\\}$ such that $B_{1}=\langle M\rangle_{FG}$ and
$m\leq r$. Thus

$\dim_{F}(B)\leq\dim_{F}((FG)a_{1})+\cdots+\dim_{F}((FG)a_{m})\leq m|G|\leq
r|G|.$

If $0\not=d\in A\setminus B$, then $B_{2}:=B_{1}+(FG)d$ is a f.g. module, such
that $B_{1}\leq B_{2}$ but $B_{1}\not=B_{2}$. Similarly, continuing this
process, we construct a family of proper submodules $\\{B_{k}<A\mid
k\in\mathbb{N}\\}$ such that each $\dim_{F}(B_{k})\leq r|G|$. Consequently,
there exists $m\in\mathbb{N}$ such that $B_{m}=A$, so $A$ is a f.g. module and
$\dim_{F}(A)\leq r|G|$.∎

The intersection of all maximal $R$-submodules of an $R$-module $A$ over a
ring $R$ is called the Frattini submodule of $A$ and denoted by
$Fratt_{R}(A)$. For finitely generated modules, the intersection of all of
maximal submodules is also called the radical of the module by analogy with
the Jacobson radical for rings. Recently, however, some other radicals have
appeared in modules, therefore we followed the example of Hartley [17] and
called the intersection of all maximal submodules the Frattini submodule.

We freely use the fact that $Fratt_{R}(A)$ consists of non-generating elements
of $A$.

######  Lemma 2.9.

Let $A=\langle M\rangle$ be an $R$-module over a ring $R$. If $S$ is a finite
subset of $M$ such that $S\subseteq Fratt_{R}(A)$, then $A=\langle M\setminus
S\rangle$.

###### Proof.

First, assume that $S=\\{d\\}$. Suppose the contrary, let the subset
$M\setminus\\{d\\}$ generate a proper $R$-submodule $B$. Clearly $d\not\in B$.
Let $\mathfrak{S}(B)$ be a family of all $R$-submodules of $A$ such that each
member of $\mathfrak{S}(B)$ includes $B$ and does not contain the element $d$.
Clearly, $B\subseteq\mathfrak{S}(B)\not=\emptyset$ and $\mathfrak{S}(B)$ has a
maximal element $C$ by Zorn’s lemma. If $D$ is an $R$-submodule of $A$ such
that $C\leq D$ and $C\not=D$, then $d\in D$. However, in this case $M\leq D$,
which implies that $D=A$. Consequently, $C$ is a maximal $R$-submodule of $A$.
On the other hand, $d\not\in C$, so we get a contradiction with a choice of
$d$, which proves the result. The rest follows by induction on $|S|$. ∎

######  Lemma 2.10.

Let $A$ be a Noetherian $R$-module over a ring $R$. If $M\subseteq A$ such
that $A/Fratt_{R}(A)=\langle a+Fratt_{R}(A)\mid a\in M\rangle_{R}$, then
$A=\langle M\rangle_{R}$ and

$d_{R}(A)=d_{R}(A/Fratt_{R}(A)).$

Note that, outwardly, our Lemma 2.10 is very similar to the following
consequence of the Nakayama’s lemma [12, Corollary 4.8(b), p. 124]. In the
Nakayama’s lemma [12, Corollary 4.8(b), p. 124], was considered a submodule,
obtained by acting on the entire module by the radical of the ring. However in
our Lemma 2.10 we consider a factor-module by the radical of the module, i.e.
at the intersection of all maximal submodules. We provide the following simple
example:

Let $A$ be a cyclic group of order $p>2$ which is considered as a module over
the ring of integers $\mathbb{Z}$. The radical (or the Frattini submodule)
$Fratt_{\mathbb{Z}}(A)$ of the module $A$ is a subgroup of order $p$ and the
radical $\mathcal{J}(\mathbb{Z})$ of the ring of integers $\mathbb{Z}$ is
zero. Consequently, $\mathcal{J}(\mathbb{Z})A=0$.

###### Proof of Lemma 2.10.

Let $B=\langle M\rangle_{R}\leq_{R}A$ such that $A\not=B$. The choice of $M$
shows that $A=B+D$. Let $B_{1}\leq_{R}A$ be generated by $M\cup\\{a_{1}\\}$,
where $a_{1}\in D\setminus B$ and $B_{1}\not=A$. Set $B_{2}=\langle
M\cup\\{a_{1}\\}\cup\\{a_{2}\\}\rangle_{R}$ such that $a_{2}\in D\setminus
B_{1}$. If $B_{2}\not=A$, then we repeat arguments above. This process can not
be infinite because $A$ is a Noetherian $R$-module, so there exist
$a_{1},\ldots,a_{n}\in D$ such that $\langle
M\cup\\{a_{1},\ldots,a_{n}\\}\rangle_{R}=A$ and $A=\langle M\rangle_{R}$.

Let $M=\\{a_{1},\ldots,a_{k}\\}\subseteq A$, such that $k\in\mathbb{N}$ is
minimal with the properties $A=\langle M\rangle_{R}$ and $k=d_{R}(A)$.
Clearly, $A/D=\langle a_{1}+D,\ldots,a_{k}+D\rangle_{R}$, where
$D=Fratt_{FG}(A)$. It follows that $d_{R}(A/D)\leq d_{R}(A)$. On the other
hand, let $\\{x_{1},\ldots,x_{m}\\}$ be a finite subset of $A$ such that
$A/D=\langle x_{1}+D,\ldots,x_{m}+D\rangle_{R}$. Thus $A=\langle
x_{1},\ldots,x_{m}\rangle_{R}$. Consequently, $m\geq d_{R}(A)$ and
$d_{R}(A/D)\geq d_{R}(A)$, so that $d_{R}(A/D)=d_{R}(A)$. ∎

######  Lemma 2.11.

Let $FG$ be the group algebra of a finite group $G$ over a field $F$. Let $A$
be a f.g. $FG$-module. If $d_{FG}(A)=k$, then $A/Fratt_{FG}(A)$ is semisimple
of finite width. Moreover, each homogeneous component has width at most $k$
and there exists a homogeneous component, whose width is exactly $k$.

###### Proof.

Let $M=\\{a_{1},\ldots,a_{k}\\}\subseteq A$ such that $k\in\mathbb{N}$ is
minimal with the property $A=\langle M\rangle_{FG}$. Clearly, each $\langle
a_{j}\rangle_{FG}\cong FG/Ann_{FG}^{left}(a_{j})$,
$\dim_{F}((FG)a_{j})\leq|G|$ and

$\dim_{F}(A)=\dim_{F}(FGa_{1}+\cdots+FGa_{k})\leq k|G|.$

Since $\dim_{F}(A)$ is finite, $A$ is a Noetherian $FG$-module. Let
$\\{V_{1},\ldots,V_{s}\\}$ be a collection of maximal $FG$-submodules of $A$.
Let $V=V_{1}\cap\cdots\cap V_{s}$. Using the Remak’s theorem we obtain an
embedding of $A/V$ into a direct sum $A/V_{1}\oplus\cdots\oplus A/V_{s}$, in
which each factor-module $A/V_{j}$ is simple. Since each submodule of a
semisimple module is itself semisimple (see [19, Corollary 4.4]), $A/V$ is a
semisimple $FG$-module. It follows that $Fratt_{FG}(A)\leq V$.

Let $0\not=b\in V\setminus Fratt_{FG}(A)$. Obviously, $A$ contains a maximal
$FG$-submodule $W$ such that $b\not\in W$. Put $U=V\cap W$, then A/U is
embedded into $A/V\oplus A/W$. Using the above arguments, we obtain that $A/U$
is a semisimple $FG$-module, so that again $Fratt_{FG}(A)\leq U$. In addition,
the choice of $U$ implies that $U\not=V$, therefore $\dim_{F}(U)<\dim_{F}(V)$.
If $Fratt_{FG}(A)\not=U$, then we repeat the above argument. This process
should be finite, because $\dim_{F}(A)$ is finite. Consequently,
$A/Fratt_{FG}(A)$ is semisimple of finite width.

Finally $d_{R}(A/Fratt_{FG}(A))=d_{R}(A)=k$ by Lemma 2.10 and use Corollary
2.7. ∎

Consider now the case when a Sylow $p$-subgroup of $G$ is normal.

######  Lemma 2.12.

Let $FG$ be the group algebra of a finite $p$-group $G$ over a field $F$ of
prime characteristic $p$. The length of a $G$-central series of an $FG$-module
$A$ is finite and bounded by $|G|$.

###### Proof.

A semidirect product $A\rtimes G$ is a nilpotent group (see [1, Lemma 3.8, p.
228]). In particular, $A\rtimes\langle g\rangle$ is also nilpotent for each
element $g\in G$.

Set $A_{0}:=A$ and $A_{i}:=(g-1)^{i}A$ for $i\geq 1$. It is easy to see that

$\begin{split}A_{k}&\not=\langle
0\rangle,\quad(g-1)A_{k}\not=A_{k},\qquad\qquad(0\leq k\leq|g|-1)\\\
A_{|g|}&=(g-1)^{|g|}A=(g^{|g|}-1)A=\langle 0\rangle.\end{split}$

Now we use induction on $|G|$. If $|G|=p$, then $G=\langle g\rangle$ is cyclic
and

(1) $A=A_{0}\geq A_{1}\geq\cdots\geq A_{|g|}=\langle 0\rangle$

is a $G$-central series.

Assume now $|G|>p$. Since a finite $p$-group is nilpotent,
$\zeta(G)\not=\langle 1\rangle$ and there exists $1\not=g\in\zeta(G)$ such
that the series (1) is $\langle g\rangle$-central, so that
$g\in\mathrm{C}_{G}(A_{j}/A_{j+1})$, where $\mathrm{C}_{G}(X)$ is the
centralizer of a set $X$ in $G$. An inclusion $g\in\zeta(G)$ shows that each
$A_{j}$ is an $FG$-submodule of $A$. Consequently, each factor $A_{j}/A_{j+1}$
can be considered as an $F(G/\langle g\rangle)$-module. Using the inequality
$|G/\langle g\rangle|<|G|$ (because $g\not=1$), we can apply the induction
hypothesis for $A_{j}/A_{j+1}$. By the induction hypothesis, $A_{j}/A_{j+1}$
has a $(G/\langle g\rangle)$-central series whose length is at most
$|G/\langle g\rangle|$. Obviously, factors of such series are also $G$-central
for each $0\leq j\leq k-1$. Thus $A$ has a $G$-central series whose length is
at most $|g||G/\langle g\rangle|=|G|$. ∎

##  3\. Proofs

###### Proof of Theorem 1.

Let $A$ be an $FG$-module in which $FG$ is the group algebra of a finite group
$G$ over the field $F$. Since $A$ has finite $r$-generator property, $A$ has
finite $F$-dimension by Lemma 2.8, so $A$ has a finite series of
$FG$-submodules, whose factors are semisimple. Let $B\leq_{FG}A$ such that
$d_{FG}(B)=r$ and $D=Fratt_{FG}(B)$. Corollary 2.10 implies that
$d_{FG}(B)=d_{FG}(B/D)=r$ and the factor-module $B/D$ is semisimple and it has
finite width by Lemma 2.11. Moreover, every homogeneous component has width at
most $r$ and there exists a homogeneous component, whose width is exactly $r$.

Let $V/W$ be a semisimple homogeneous $FG$-factor and set $U:=Fratt_{FG}(V)$.
Obviously, $U\leq W$ and the inequality $d_{FG}(V)\leq r$ implies that
$d_{FG}(V/U)\leq r$ by Corollary 2.10. Consequently, $V/U$ is semisimple by
Lemma 2.11 and each of its homogeneous components has width at most $r$. The
isomorphism

$V/W\cong_{FG}(V/U)/(W/U)$

implies that $V/W$ has width at most $r$.

Conversely, let $C/E$ be a semisimple homogeneous $FG$-factor, having width
$r$. Thus $Fratt_{FG}(C)\leq E$, so $C/Fratt_{FG}(C)$ has a homogeneous factor
whose width is exactly $r$. By the statements of our theorem, each another
homogeneous factor of $C/Fratt_{FG}(C)$ has width at most $r$. Then
$d_{FG}(C/Fratt_{FG}(C))=r$ by Lemma 2.11, and $d_{FG}(C)=r$ by Corollary
2.10. Finally, let $S$ be an $FG$-submodule of $A$. The factor
$S/Fratt_{FG}(S)$ is semisimple by Lemma 2.11 and each homogeneous factor of
$S/Fratt_{FG}(S)$ has width at most $r$ by parts (i)-(ii) of our theorem.
Finally, using Lemma 2.11, we obtain that $d_{FG}(S/Fratt_{FG}(S))\leq r$ and
$d_{FG}(S)\leq r$, so $A$ has $r$-generator property. ∎

###### Proof of Theorem 2.

Let $F$ be a field of prime characteristic $p$ and let $G$ be a finite group,
such that $p$ does not divide $|G|$. If $A$ is an $FG$-module, then $A$ is
semisimple (see [19, Corollary 5.15]). It follows that the group ring $FG$ is
semisimple. Denote by $nns_{F}(G)$ the number of pairwise non-isomorphic
simple $FG$-modules. This number is finite (see [2, 3]) and can be calculated
in the following way.

Let $\xi$ be a primitive $n$-th root of unity, where $n$ is the greatest
divisor of $\exp(G)$ which is not divisible by $char(F)$. Two elements $a,b\in
G$ are called $F$-conjugate if there exist $x\in G$ and $m\in\mathbb{N}$ such
that $x^{-1}ax=b^{m}$ and the map $\xi\mapsto\xi^{m}$ is extensible to an
automorphism of the field $F(\xi)$ fixing the subfield $F$ elementwise. Berman
[3] and Witt [24] proved that $\mathrm{nns}_{F}(G)$ is equal to the number of
$F$-conjugate classes of $F$-regular elements of $G$ (see [9, Theorem 42.8 p.
306]). If $F$ is algebraically closed, then Brauer [6, 9] shows that
$\mathrm{nns}_{F}(G)$ is equal to the number of the conjugate classes of
elements of $G$ (see [9, Theorem 40.1 p. 283]). In both cases
$\mathrm{nns}_{F}(G)$ is bounded by the number of conjugacy classes of $G$.

Since $\dim_{F}(A)$ is finite, $A=\oplus_{j=1}^{n}H_{j}$ in which each $H_{j}$
is a homogeneous component of $A$. As we noted above $n\leq nns_{F}(G)$. The
fact that $H_{j}$ is a direct sum of at most $r$ simple submodules follows
from Corollary 2.7(i-iii). ∎

###### Proof of Theorem 3.

Clearly, $\dim_{F}(A)$ is finite by Lemma 2.8 and $A$ has a series of
$FG$-submodules (see Lemma 2.12)

$\langle 0\rangle=Z_{0}\leq Z_{1}\leq\cdots\leq Z_{m-1}\leq Z_{m}=A$

whose factors are $p$-central, that is $P\leq\mathrm{C}_{G}(Z_{j+1}/Z_{j})$
for each $0\leq j\leq m-1$. It follows that each
$G/\mathrm{C}_{G}(Z_{j+1}/Z_{j})$ is a $p$-group, so we can apply Theorem 2 to
each of these factors. ∎

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