Questions tagged [socle]

Questions relating to the direct product of minimal normal subgroups in a group or sum of all minimal nonzero submodules of a module.

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Calculation of the socle of $R$ for given algebra $R$

I am thinking of calculation $Soc(R)$ for given algebra $R$. But the definition of socle is too abstract to calculate. https://en.m.wikipedia.org/wiki/Socle_(mathematics) Does anyone tell me how to ...
masa's user avatar
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The left socle and right socle of a semiprime ring coincide.

Let $R$ be a semiprime ring. Then the right socle and left socle coincide. (The right/left socle for a ring is the sum over all minimal right/left ideals of $R$.) You can assume that the left/right ...
Laak's user avatar
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Socle of a group and the Fitting subgroup of a solvable primitive group [closed]

I have been puzzled by this lately. And first, thank you to whoever takes the times to read through these questions. Consider a finite group $G$ which is solvable. Let $F(G)$ define the Fitting ...
Claudio Piedade's user avatar
2 votes
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Socle series and radical series have same length

I have been given a module $V$ of finite length $l(V)=n$. If I am not wrong if follows from Jordan- Hölder theorem that the socle series and radical series have also finite length since they are ...
Mikel Martinez Puente's user avatar
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If $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$, then $k$ is not a direct summand of $N$?

Let $M$ be a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$...
feder's user avatar
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$\text{soc}(V)$ and $\text{rad}(V)$ of a representation $V$ of an acyclic quiver $Q$

Let $Q$ be an acyclic quiver, and $V = (V_i, V_a)$ a representation of $Q$. I want to show that $$\text{soc}(V) = \bigoplus_{i \in Q_0} \left(\bigcap_{a \in Q_1, s(a) = i} \ker(V_a) \right), \hspace{...
LinearAlgebruh's user avatar
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When is the intersection of every non-zero ideal with socle again non-zero?

Let $(R,\mathfrak m)$ be a Noetherian local ring such that $(0:_R \mathfrak m)\neq 0$. Then, is it true that for every non-zero ideal $I$ of $R$, one also has $I\cap (0:_R \mathfrak m)\neq 0$ ? I know ...
feder's user avatar
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${\rm Soc}(G) $ simple implies $ G $ almost simple

Recall that a group $ G $ is called almost simple if there exists a (non-abelian) simple group $ S $ such that $ S \leq G \leq {\rm Aut}(S) $. If $ G $ is almost simple then $ {\rm Soc}(G) $, the ...
Ian Gershon Teixeira's user avatar
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What is socle friendly group means?

In the paper by Ben Elias et al. (2010), the authors has defined the socle friendly group. Definition 1: Let $G$ be an arbitrary finite group, and let $\mathcal{T}=\{T \triangleleft G \,|\, T \leq \...
Jins's user avatar
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If $G$ is a finite $p$-group, then the socle of $G$ is contained in the centre $Z(G)$

This is an exercise in Permutation Group by D.Dixon and Brian Mortimer, from page112, exercise 4.3.3. Recall: The socle of $G$ is the subgroup generated by the set of all minimal normal subgroups of $...
Zihao huang's user avatar
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Given an $R$-module $M$, then $soc(M) = ann_M(J(R))$ if $R/J(R)$ is semisimple

Given an $R$-module $M$: $J(R)$ is the Jacobson radical of $R$. Given $X \subset R$, $ann_M(X) = \{m \in M: \forall r \in X, rm = 0\}$. $Soc(M)$ is the socle of $M$ (the submodule of $M$ generated ...
Diogo Santos's user avatar
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Minimal normal subgroups of normal closure

This question arose from my attempt at understanding the answer in this post - the comments below it, to be precise. Everything revolves around the following problem: Let $S$ be a simple, nonabelian ...
Gauss's user avatar
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Simple subgroups and the Wielandt subgroup

My goal is to prove the following: Let $G$ be a finite group and let $S \leq G$ be a simple subgroup. Suppose that $SH = HS$ for all subnormal subgroups $H$ of $G$. Show that $S$ is contained in the ...
Gauss's user avatar
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Socle of a Hamiltonian group?

Currently I am studying the definition of Socle of a group. Also, I came to know that Socle of a finite nilpotent group $G$ is the product of elementary abelian $p$-groups for the collection of primes ...
Debarati's user avatar
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Index of Socle of primitive permutation group $G$

This is a follow up question of this (Edit: The following question is for primitive group $G$ which lies in the case (i) and does not lie in case (ii) of Theorem 5.6C of the textbook "Finite ...
Jins's user avatar
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Is the Socle of an almost simple group a simple group?

Let $G$ be a finite primitive group of degree $n$, and let $H$ be the socle of $G$. Then if $H$ is isomorphic to a direct power $T^m$ of a nonabelian simple group $T$ then the following holds when $m=...
Jins's user avatar
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Doubt with Socle and O'Nan-Scott Theorem.

The following is the statement of O'Nan-Scott Theorem. Theorem: Let $G$ be a finite primitive group of degree $n$, and let $H$ be the socle of G. Then either (a) $H$ is a regular elementary abelian $p$...
Jins's user avatar
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Identify the socle of an abelian group.

This is Exercise 3.3.11 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. The Details: Since definitions vary, on ...
Shaun's user avatar
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Let $G$ be a finite group. Then $G$ is completely reducible iff $G={\rm Soc}(G)$.

This is Exercise 3.3.10 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE, although the equivalence in question is mentioned here. ...
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Showing that if $G$ is a solvable finite primitive group and $M$ is a core-free maximal subgroup of $G$, then $O_p(M)=1$ where $p$ divides $|Soc(G)|$

I’m trying to show that if $G$ is a solvable finite primitive group and $M$ is a core-free maximal subgroup of $G$, then $O_p(M)=1$ where $p$ divides $|Soc(G)|$ ($Soc(G)$ is the socle of $G$). I know ...
dahemar's user avatar
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When is commutative reduced ring a finite direct product of domains?

All rings considered are unital and commutative. Intro Direct products of domains are reduced rings. The opposite is not true but I must admit I have troubles finding counter-examples. It holds that ...
dmk's user avatar
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Equivalent Definitions of the Socle of a Module

Wikipedia gives the following definitions of the socle of an $R$-module $M$: $$\text{Soc}(M)=\sum \left\{S:S\subseteq M\text{ is simple}\right\}:=S_1$$ and $$\text{Soc}(M)=\bigcap\left\{ E:E\subseteq ...
Dave's user avatar
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Is Socle of evey locally Noetherian module a direct summand

Definition: An $R$-module $M$ is called locally Noetherian if, any finitely generated submodule of $M$ is Noetherian. Question: Let $R$ be a ring with identity and $M$ be a locally Noetherian $R$-...
bipin's user avatar
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if $U$ is not projective, then $U_1^P$ is a proper subspace of $U^P$

Suppose that $k$ is a field of prime characteristic $p$. Let $P$ be a finite $p$-group, and let $U$ be a finitely generated $kP$-module. Then: (i) We have $soc(U) = U^P$ ($P$ fixed points in $U$). (ii)...
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Must $\operatorname{Soc}(R)^2 =\operatorname{Soc}(R)^3$ (for a ring $R$)?

I am struggling with the following problem. Show that the right socle of a ring $E := \operatorname{Soc}(R)$ has $E^2 = E^3$. I know that $E$ is a two sided ideal and so $E^3 \subset E^2$. I am also ...
James's user avatar
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Socle length via semisimple filtration

I've been working through Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and I've gotten quite stuck on exercise II.6, which involves not-necessarily-Artin rings. The exercise ...
Will Dana's user avatar
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socle and essential extension of finitely generated mod over an artin algebra

In page 40 of Auslander‘s representation theory of artin algebra, the proposition 4.1, For $A$ in mod$\Lambda$ where $\Lambda$ is an artin algebra we have the following. a) $A=0$ iff $socA=0$ ...
subHangLou's user avatar
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Socle of representations of Klein Four group over $\mathbb{F}_2$

I am trying to answer Q7. from page 11 of http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf The question states: "Let $G= \langle x, y \ \mid \ x^2=y^2=1 \rangle$ be the Klein four-group,...
robotsheepboy's user avatar
1 vote
1 answer
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Confusion over Socle

So, I can write down the definition of the socle of a module but I'm having trouble actually putting it in my brain. To understand things I'm trying to write down the radical and socle series of the ...
RhythmInk's user avatar
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Infinite socle series

Is it possible for a finite dimensional $k$-algebra $A$ ($k$ an algebraically closed field of characteristic $p$) to have an infinite socle series? Namely, I have calculated the radical and socle for ...
RhythmInk's user avatar
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6 votes
1 answer
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Socle of a direct product of finite groups.

The socle of a group $G$ is defined as the subgroup generated by minimal subgroups among normal subgroups of $G$, and it is denoted as $\textrm{Soc}(G)$. Suppose $A_1,...,A_n$ are finite groups. Is ...
John P's user avatar
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Definition of socle of a module

For me, the socle of an $R$-module $M$ is the unique maximal semisimple submodule of $M$. If $(R,\mathfrak m)$ is a local ring, this is equivalent to the maximal submodule annihilated by $\mathfrak m$....
Bubaya's user avatar
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A $\mathfrak{sl}_2(\mathbb{C})$-module $M$ is torsion free iff $\text{soc}(M)$ is torsion-free

Let $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$. I'm reading a proof that a $\mathfrak{g}$-module is torsion-free iff its socle is torsion-free. One direction is clear, but for the other, if $M$ has ...
Hailie Mathieson's user avatar
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1 answer
975 views

The radical and socle series of a module and its dual

I am study Peter Webb's book "A course in finite group representation theory", and stuck on ex.7 of chapter 6. The exercise is about the relationship between the socle series and radical series of a ...
Hsin-Chieh Liao's user avatar
1 vote
1 answer
141 views

Decomposition of socle of a ring

My question is divided by two: "(1) Is the right socle $S_r$ of an arbitrary unital ring $R$ is decomposed as $S_r=S_1\bigoplus S_2$, where $S_1$ is the sum of all the nilpotent minimal right ideals, ...
karparvar's user avatar
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1 vote
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Determine $Soc(R_R)$ for $R =\left(\begin{smallmatrix} K & K \\ 0 & K \end{smallmatrix}\right)$

Let me continue the study I began in a previous post on the ring of upper triangular matrices: We consider $K$ a field and the ring $R = \begin{pmatrix} K & K \\ 0 & K \end{pmatrix}$. Recall ...
Jacques Saliba's user avatar
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Socle of a quotient ring

I want to know the form of the left socle of a factor ring $R/I$, where $I$ is an ideal of a unital ring $R$. It is, by definition, the sum of all simple left ideals of the ring $R/I$. So, one should ...
karparvar's user avatar
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socle of a quotient module

As the title says, I am interested on the socle of a quotient module. Let $R$ be a ring and $M,N$ two left $R$-modules. If $f: M \rightarrow N$ then $f(socle(M)) \subseteq socle(N)$. In particular, if ...
user 1987's user avatar
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Socle of a semi-local ring

Let $R$ be a commutative ring with identity. Suppose that $R$ is semi-local with maximal ideals $P_1,...,P_n$. By the Chinese Remainder Theorem we have $$R/J(R) \simeq R/P_1 \times ... \times R/P_n$$ ...
Shimrod's user avatar
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1 answer
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The socle of an almost simple group

By definition an almost simple group is a group $G$ with $$T \cong \mathrm{Inn}(T)\trianglelefteq G \leq \mathrm{Aut}(T),$$ where $T$ is a non-abelian simple group. How would one show that in this ...
Ishika's user avatar
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Normal closure is minimal normal subgroup

The following problem is from the book "Finite Group Theory" by Martin Isaacs. (2.A.7) Let $S \lhd \lhd G$ (S is subnormal in G), where $S$ is nonabelian and simple and $G$ is finite. Show ...
Stefan4024's user avatar
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Artinian module's socle is essential

How do I prove that the socle of an Artinian module is an essential submodule? I don't see where we should use the artinianity of the module to prove this.
Nesa's user avatar
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Finding a form of Representations of algebra $FV_4$, where $F=Z_2$

I am hoping for some help with the following exam question I attempted. It is as follows, broken in to two parts: Let $A$ be a finite-dimensional algebra over a field $F$. $1.)a)i)$ What is a ...
SEWillB's user avatar
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1 answer
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A subring of the upper triangular matrices

I want to compute the Jacobson radical, the right socle, and the left socle of the ring of $3\times 3$ upper triangular matrices with entries in $\mathbb Z_4$ and such that the entries on the main ...
karparvar's user avatar
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1 vote
1 answer
443 views

Whether a non-zero module can have a zero socle?

Let $M$ be a module. Then $soc(M)=\sum\{N\leq M| \text{$N$ is a simple submodule of $M$}\}=\cap\{L\leq M| \text{$L$ is essential in $M$}\} $. I don't know whether a non-zero module can have zero socle?...
Daisy's user avatar
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Is this proof that Soc(DM) = DTop(M) correct?

I was struggling with this problem regarding the socle and the top of a finitely generated right-module $M$ over a finite dimensional $K$- algebra $A$. If you define the duality functor $DM = Hom_{k}(...
Werner Germán Busch's user avatar
3 votes
1 answer
194 views

A prime ring whose socle is nonzero and of finite length is simple Artinian.

This is a part of an exercise (Sect. 14 Exercise 11) in Anderson & Fuller's book "Rings and Categories of Modules", and I'm a graduate level student in Turkey. I want to prove that if $R$ is a ...
akivura's user avatar
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1 answer
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Socle degree of Artinian ring

I started to study properties of Artinian and Gorestein Rings, trying to approach the Fröberg conjecture, but I note that I am having trouble computing some examples. For instance, I would like to ...
User43029's user avatar
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socle of polynomial rings

I want to know what the "one-sided" (say, left)) ideals of the polynomial ring $R[x]$ are, as to obtain the left socle of this ring. If $I$ is a left ideal of $R$, then $I[x]$ comprising all ...
karparvar's user avatar
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1 vote
2 answers
74 views

Searching for a certain local ring

I am searching for a local ring $R$ such that $R/\mathrm{Soc}(R_R)$ is not isomorphic to the two-element ring $\mathbb Z_2$. Here, by $\mathrm{Soc}(R_R)$ I mean the socle of the right $R$-module $R$....
karparvar's user avatar
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