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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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 $G= \langle x, y \ | \ x^2=y^2=1 \rangle$ be the Klein four-group, $...
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30 views

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 ...
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30 views

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 ...
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106 views

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 $\mathfrak m$....
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58 views

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 ...
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161 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 ...
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81 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, ...
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83 views

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 ...
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107 views

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 ...
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80 views

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 ...
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97 views

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 $$R/J(R) \simeq R/P_1 \times ... \times R/P_n$$ ...
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218 views

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 ...
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150 views

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 that $S^...
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138 views

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.
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117 views

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 ...
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201 views

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 ...
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133 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?...
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55 views

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}(...
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78 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 ...
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150 views

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 ...
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83 views

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 ...
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64 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 $R$....
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90 views

Direct product of socles

If $R$ and $S$ are rings with identity, is it true that socle is preserved under direct product, namely, is it true that $\mathrm{Soc}(R)\times \mathrm{Soc}(S)=\mathrm{Soc}(R\times S)$? (I mean by $\...
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30 views

Decomposition of right ideals

Let $R$ be a unital ring with right socle $Soc(R_R)$ such that $R/Soc(R_R)$ is right weakly regular, i.e all whose right ideals are idempotent. Is it true that every right ideal $I$ of $R$ decomposes ...
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153 views

Question concerning cyclic modules and modules where the socle of $M$ is irreducible

Let $R$ be a ring and $M$ be an $R$-module of finite length $n$ (given by the number of composition factors). I am trying to prove or disprove: If $\mathrm{Soc}(M)$ (the socle of $M$) is a unique ...
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151 views

Socle of a primitive permutation group

So we need to prove that the socle of a primitive permutation group is a direct product of isomorphic simple groups. Now socle means product of the minimal normal subgroups. I know that every non-...
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175 views

A question about the socle of an algebra.

Let $A$ be a finite dimensional algebra over a field K. Suppose $e$ is an idempotent such that $Ae$ is the direct sum of all indecomposable injective-projective left $A$-modules. Is that $soc(_AA) \in ...
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77 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
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229 views

the socle is semisimple

The socle of a module $M$ is semisimple. I know that the socle is $soc(M)=\sum\{N\leq M| \text{$N$ is a simple submodule of $M$}\}$. So $soc(M)=N_1+\cdots+N_k$, where $N_j$ are simple submodules, for ...
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63 views

Show that Soc($\bigoplus_{\alpha\in I}M_\alpha$)=$\bigoplus_{\alpha\in I}\text{Soc}(M_\alpha)$

Given left $R$-modules $\{M_\alpha\}_{\alpha\in I}$ for some index set $I$, I'd like to show that the socle of their direct sum is the direct sum of each module's socle, that is, $$\text{Soc}(\...
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90 views

socle(M) being simple gives an upper bound for the dimension of End(M)?

Suppose $k$ is a algebraically closed field of arbitrary characteristic. Let $A$ be a finite dimensional $k$-algebra and $M$ an $A$-module with finite dimension with respect to $k$. I have seen it ...
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114 views

Why there is an isomorphism $D(soc^{i}M) \cong DM/rad^{i}DM$

I am reading the book Elements of the Representation Theory of Associative Algebras, volume 1, by Assem et al. On page 162, it is written $D(soc^{i}M)\cong DM/rad^{i}DM$, where $DM=\mathrm{Hom}(M,k)$. ...
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259 views

The property of a module that has a simple socle

Let M be a module that has a simple socle.I can get that M is indecomposable and all submodules of M contain socM. Are there any other properties of M? Can we character the structure of M? And is M/...
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72 views

(Reference Request) Socle of a module.

I came across the term "Socle" of a module defined for a finitely generated module $M$ over a noetherian ring $(A,\mathbb{m})$ as follows $$\mathrm{Soc}(M) = \lbrace x \in m \mid ax = 0\ \forall a \...
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38 views

Show that in ascending Loewy series, $S^r(R)=R$

Let $R$ be an Artinian ring, $N$ its radical, and $r$ the smallest natural number such that $N^r=0$. Define an ideal $S^n(R)$ of $R$ recursively as follows: $S^1(R)=soc(R)$ Assuming $S^i(R)$...
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241 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then $(a+b)x=ax+bx$. ...
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453 views

Socle of a ring $R$

It is well-known that for an idempotent $e\in R$, the right $R$-module $eR$ is simple faithful if and only if $Re$ is a simple faithful left $R$-module. Now, I want to prove that when $Re$ is ...
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262 views

Canonical Module and Socle of an Artinian $k$-Algebra

Let $R$ be an Artinian $k$-algebra generated by elements of degree $1$. Denote the canonical module of $R$ by $\omega_R$. By Theorem 3.6.19 in Bruns and Herzog (CMR), we have that $\omega_R = (H_m^d(...
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179 views

Socle of submodule relative to the module

in these notes i am reading i am told that the socle of $K$ (where $K \subset M$ , and $M$ is a module) is = $K \cap$ Soc $ M$ But why is this? i see the intuition but cannot formalize a proof any ...
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246 views

Showing the socle of a module exists and is unique

I have been set the following exercise: For every $R$-module $M$, show that there exists a unique semi-simple submodule $sM$ $\subset$ $M$ which contains every semi-simple submodule of $M$. After ...
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277 views

On one-dimensional socles

Let $(R,m,k)$ be a regular local ring of dimension $n$. Let $b_1,\dots,b_n$ be a maximal $R$-sequence and define $J=(b_1,\dots,b_n)$. Let $y_1,\dots,y_n$ be a regular system of parameters of $R$ and ...
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60 views

If a group 's socle is $Sz(q)$, how can I determine its maximal subgroup

As the title says,I want to know the maximal subgroup of the group whose socle is $Sz(q)$. If there are such papers,could you tell its name or give me a link. Thank a lot
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259 views

How to understand the automorphism group of a very symmetric graph (related to sylow intersections)

For a group $G$ and subgroup $H$, consider the relation on $G$ defined $x \sim y$ if $H^x \cap H^y = 1$. This defines a graph on $G$. It is always fairly symmetric: $N_G(H)$ acts on the left and $G$ ...
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151 views

Socle of abelian divisible periodic group

I'm trying to prove that the socle of a periodic divisible abelian group J is a proper subgroup of J. I know that J is direct sum of quasicyclic groups, say $$ J={\oplus}_{i\in I} P_i $$ and that ...
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180 views

Questions about socles and radicals.

Let $M$ be a module of a finite dimensional algebra $A$ over an algebraically closed field $K$. Let $N=M/\operatorname{rad}M$ be the top of $M$ and suppose that $N$ is simple. Let $D=\hom_K(\cdot, K)$....
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310 views

socles of semiperfect rings

For readers' benefit, a few definitions for a ring $R$. The left (right) socle of $R$ is the sum of all minimal left (right) ideals of $R$. It may happen that it is zero if no minimals exist. A ring ...
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140 views

Calculation of dimension of Socle

Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $$\dim_k \operatorname{Soc}(S/t^3S)=2?$$.($\dim_k$ means dimension as $k$-vector space.) background: $\operatorname{...
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127 views

Infinite normal subgroup - Socle

I am looking for an infinite normal subgroup s.t. for $N \unlhd G$ we have $N \cap Soc(G) \not\subset G$. It must be infinite, since we proved in the lecture, that for finite $N$, $N \cap Soc(G)$ is ...
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129 views

normal subgroup + Socle

I am looking for examples of socle and normal-subgroup relations. If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) ...
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239 views

Maximal subgroups of almost simple groups with socle $PSL(2, q)$

Let $G$ be an almost simple group with socle $PSL(2,q)$ where $q=p^f>3$ is the $f$th power of some odd prime $p$, and $M$ a maximal subgroup of $G$. By http://arxiv.org/abs/math/0703685, except for ...