# 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 ...
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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 ...
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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 ...
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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 ...
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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 $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$...
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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 $R$-...
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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 ...
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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 ...
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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 $socA=0$ ...
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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,...
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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 ...
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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 ...
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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 ...
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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 $\mathfrak m$....
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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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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.
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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 ...
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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 ...
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### 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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