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Questions tagged [semi-simple-rings]

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2answers
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An infinite product of fields is not a semisimple ring

I want to show that an infinite product of fields is not a semisimple ring. I know Artin-Wedderburn Theorem, but I wonder can I explain it without using this theorem? Any help will be appreciated. ...
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0answers
5 views

Maximal separable subalgebras of semisimple associative algebras

Is anything known about maximal separable subalgebras of semisimple associative (and not necessarily commutative) algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic ...
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0answers
39 views

Group ring $R[G]$ semisimple if and only if $J = J^2$?

Let $G$ be an abelian group and let $R$ be a commutative ring and consider the group ring $R[G]$ of finite formal linear combinations of elements of $G$ with coefficients in $R$. Let $J = (1 - g ~|~ g ...
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0answers
17 views

Simple ring that is not semi-simple [duplicate]

Can someone give me an example of a simple ring that is not semi-simple ?
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2answers
44 views

Is $K[X]$ not a semilocal ring?

Let $K$ be a field. We will write $K[X]$ to denote the set of all polynomials in one variable over the field $K$ and $\mathrm{Maxspec}(K[X])$, the set of all maximal ideals of $K[X].$ Also, we call a ...
2
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1answer
50 views

For an Artinian ring semiprimitive implies semisimple.

I'm currently reading Rotman's An Introduction to Homological Algebra (2nd edition), and on page 188 in the proof of Theorem 4.66 (Every left Artinian ring is semiperfect), I ran across the claim: ...
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1answer
16 views

Reduced rings and simple modules

I'm having trouble getting my intuition pumping and the details in my head on the Jacobson radical (intersection of maximal ideals or intersection of annihilators of simple modules). What I'm ...
2
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2answers
339 views

Show that this ring is semi-simple

Let $\mathbb{F}_q$ be a field of order $q = p^m$, where $p$ is the characteristic of the field; a prime. Consider the ring $$R_n = \mathbb{F}_q[x]/\langle x^n - 1 \rangle $$ Now, I've read that this ...
1
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1answer
33 views

Two simple modules over a semisimple ring are isomorphic when their annihilators are equal

We know that if $M_R$ and $N_R$ are isomorphic $R$-modules, then their annihilators $ann(M) = ann(N)$. The converse is not true in general. But if $R$ is a semisimple ring and $M_R$ and $N_R$ are ...
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2answers
45 views

$R$ semisimple $\implies IJ = I \cap J$

Let $R$ be a semisimple ring, $I$ be a right ideal of $R$ and $J$ be a left ideal. We wish to show $IJ = I \cap J$. Clearly $IJ \subseteq I \cap J$. Since $R$ is semisimple and $J$ is a left $R$-...
2
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1answer
30 views

$R$ semisimple $\implies M_n(R)$ is semsimple

I'm working out of TY Lam's first course in noncommutative rings. I'd like to show that if $R$ is a semismiple ring, $M_n(R)$ is as well. The obvious answer to me seems to be that since $R$ is a (left)...
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1answer
37 views

submodule of a semisimple module has complement

I want to prove that a submodule of a semisimple module admits a complement. That is, if S $\subset$ M, then there exists T $\subset$ M, such that M = S $\bigoplus$ T. One supposes that M = $\bigoplus$...
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1answer
31 views

$R = \mathbb{Z}/m\mathbb{Z}$, and consider $R^n$ for any $n,m$. For what values of $m$ is $R$ a semisimple $R$-module?

Let $R = \mathbb{Z}/m\mathbb{Z}$, and consider $R^n$ for any $n,m$. For what values of $m$ is $R$ a semisimple $R$-module? $\textbf{Context:}$ I am interested for what $m$ would the image of an ...
6
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2answers
77 views

Nilpotent elements of group algebra $\Bbb CG$

Goal: explicitly find a nilpotent element of the group algebra $\Bbb C G$ for some finite group $G$. This exists if and only if $G$ is non abelian by Maschke's theorem and Wedderburn-Artin. By ...
0
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1answer
33 views

Question on proof that matrix algebra over given algebra is semisimple iff original algebra is semisimple

Let $A$ be a finite-dimensional linear associative algebra over some field $F$. Then denote by $M_n(A)$ the set of $n \times n$ matrices with entries in $A$ and the usual operations. Then $M_n(A)$ ...
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1answer
36 views

Semisimple subalgebras of $M_4(\mathbb{C})$

I'm working on the following problem and I'd like some guidance. Describe up to isomorphism all semisimple $\mathbb{C}$-subalgebras of $M_4(\mathbb{C})$ (4 by 4 matrices over $\mathbb{C}$). Note ...
2
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1answer
43 views

Matrix Ring of semisimple algebra over $\mathbb{C}$

I'm working on the following question. Let $A$ be a finite dimensional semi-simple algebra over $\mathbb{C}$, and set $M_n(A)$ ring of $n$ by $n$ matrices over $A$. (a) Show that $M_2(A)$ ...
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0answers
43 views

Transpositions and idempotents in matrix algebras

Let $A$ be a semi-simple algebra over $\mathbb{C}$. An idempotent $x$ in $A$ is an element of $A$ such that $x^2 = x$. A minimal non-zero idempotent $y$ of $A$ is a non-zero idempotent of $A$ such ...
2
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2answers
38 views

Are these two isomorphisms of the group algebra with a matrix algebra distinct?

Let $k$ be an algebraically closed field of characteristic $0$ and $G$ a finite group. The representation theory of $G$ is essentially the same the as the module theory over the (non commutative) ...
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0answers
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Ring Antihomomorphisms: Understanding Isomorphisms

I'm working on an expository paper and need a little clarification. Let $R = \mathbb{M}_n(D)$ where $D$ is a division ring. Let $E_{ij}$ be the matix whose $(i,j)$-entry is 1 and all other are zeros....
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2answers
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Example of $f$ semi-simple but not diagonalizable

In our notes, we have written down: $f$ diagonalizable $\Rightarrow$ $f$ semi-simple (*) However, we also mentioned that $f$ semi-simple $\iff$ the minimal polynomial $\mu_{f}=\prod^{n}_{i=1}(X-\...
1
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1answer
49 views

On Wedderburn's theorem for $\mathbb{k}$-algebras of finite dimension with $\mathbb{k}$ algebraically closed.

I'm currently taking a first introduction to abstract algebra. At the moment, we're talking about semisimple modules and rings. We have already covered the following results Theorem (Wedderburn): $...
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1answer
147 views

Representation of direct product $G\times H$ on tensor product

Let $G$ and $H$ by finite groups. Let $V$ and $U$ be irreducible representations of $G$ and $H$, respectively. When the ground field is $\mathbb{C}$, I know how to show that $V\otimes U$ is an ...
2
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0answers
50 views

Matrix ring over a semi-simple algebra

I'm working on the following question and I'm a little stuck. Would appreciate any hints or solutions to any parts. Let $A$ be a finite-dimensional semi-simple algebra over $\mathbb{C}$ and set ...
2
votes
1answer
54 views

Question about Hopkins-Levitzki Theorem's proof

I am studying the proof of this theorem (of T.Y.Lam). Namely: Let R be a ring for which rad R is nilpotent, and $R/radR$ is semisimple. Then for any R-module ${}_{R}M$, the following statements are ...
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0answers
49 views

A question about the group algebra $k[G]$ for a finite group $G$.

Let $G$ be a finite group, $k = \mathbb C$ and consider the ring $R = k[G]$. We can think of $M = k[G]$ as a left $R$ module and by considering character theory, we can show that there is an ...
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2answers
66 views

A module $M$ whose submodules and factor modules are semisimple but not semisimple itself

An $R$-module $M$ is said to be a semisimple module if it is a direct sum of its simple submodules. I've proved the following equivalent characterization for semisimple modules: $M$ is semisimple ...
3
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2answers
53 views

Associative algebra without nilpotent ideals is direct sum of minimal left ideals

In the book 'Spinors, Clifford and Cayley Algebras' Hermann states that any (finite dimensional) semisimple associative algebra is the direct sum of minimal left ideals. Here, 'semisimple' is defined ...
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1answer
207 views

Showing commutative semisimple ring with unity is direct sum of fields

Show that if $R$ is a commutative, semisimple ring with unity then $R$ is a direct sum of fields. I have looked at the math.stackexchange post Commutative ring is semisimple iff it's isomorphic ...
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0answers
65 views

Example of a non-semiprime ring (or non right semisimple)

I would like to find a ring that is left semisimple but not right semisimple. I know that it should be non unitary, and non semiprime, but I can't find an example of a non semiprime ring, I just find ...
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2answers
41 views

A characterization of semisimple module related to anihilators

I'm trying to solve the following question: Let $R$ be a ring and let $M$ be an $R$-module. Prove that $M$ is semisimple iff $ann(m)$ is the intersection of finitely many maximal left ideals of $R$ ...
0
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1answer
75 views

A characterization for minimal left ideals of semisimple rings

I'm trying to solve the following problem: Let $R$ be a semisimple ring. Prove that a left ideal $L$ of $R$ is minimal iff it can be written as $L=Re$ where $e$ is a primitive idempotent of $R$. ...
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1answer
74 views

The ring $\mathrm{End}_D(V) $ is simple if $V$ is finite dimensional.

Theorem: Let $V$ be an $n$-dimensional vector space over a division ring $D$. Then the rings $\mathrm{End}_D(V)$ and $ M_n(D^{\mathrm{o}})$ are isomorphic. Remark: If $D$ is a division ring, then ...
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1answer
52 views

the unitization of $A$

Let $A$ be an $F$-algebra. Then the set $F \times A$ becomes an $F$-algebra, which we denote by $ A^{*}$, if we define addition, scalar multiplication and product as follows: $$(\lambda, x) + (\mu, y)...
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1answer
125 views

Simple quotient of a semisimple module is a direct summand?

I am particurarly referring to Zimmermann's Representation Theory Corollary $1.4.20$. Let $K$ be a field and $A$ a finite dimensional semisimple $K$-algebra. We have the decomposition of the regular $...
10
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1answer
345 views

A problem of central simple algebras: why $(E,s,\gamma)\cong M_n(F)$ only if $\gamma$ is the norm of an element of $E$?

I am stuck in the following problem, which is the exercise 6 of section 4.6 of N. Jacobson's Basic Algebra II: Problem. Prove that $(E,s,\gamma)\cong M_n(F)$ if and only if $\gamma$ is the norm of ...
4
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1answer
138 views

$kG\cong M_{n_1}(k)\times \cdots \times M_{n_r}(k)$ as vector spaces over $k$

I am reading Rotman's Advanced Modern Algebra. In the proof of Corollary C-2.47, the author use the fact $kG\cong M_{n_1}(k)\times \cdots \times M_{n_r}(k)$ as vector spaces over $k$. But he only ...
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1answer
31 views

An inconsistency of the definition of the simple components of a semisimple ring

The definition of the simple components of a semisimple ring which I learned is as follow (see Rotman's Advanced Modern Algebra): Let $R$ be a left semisimple ring, and let $$R=L_1\oplus \cdots \...
2
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1answer
178 views

A direct sum of semisimple left R-modules is semisimple

We define a ring has unity and semisimple module is a direct sum of simple submodules. There is a theorem states that A direct sum of semisimple left $R$-modules is semisimple. Proof: Suppose $M=...
4
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1answer
323 views

Why does the annihilator equals the kernel of $r \mapsto rx$ for simple modules in the non-commutative case

In R. Ash, Abstract Algebra in the chapter on non-commutative rings (9.2.2), the following exericse occurs: Let $M$ be a nonzero cyclic module. Show that $M$ is simple if and only if $\operatorname{...
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1answer
67 views

How to prove that the ring of upper trianglular matrices is not semisimple?

I was wondering how to prove that the ring $$R= \bigg\{ \ \left( {\begin{array}{cc} a & b \\ 0 & c \\ \end{array} } \right): \ a,b,c \in \mathbb{C} \ \bigg\} $$ is not ...
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1answer
61 views

On cyclic right $R/I$-modules for two sided ideal $I$ of $R$

Let $R$ be a ring such that every cyclic right $R$ module is either injective or projective and let $I$ be a two sided ideal of $R$ . Then why is every cyclic right $R/I$-module either injective or ...
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2answers
462 views

When a submodule N of a module M is a direct summand of M?

When a submodule N of a R-module M is a direct summand of M (or of R-module R) ?. For instance, if M is semisimple then N is semisimple, does this say that N is direct summand of M (or of R-module R)...
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1answer
80 views

End$_A(A)\cong A$ [duplicate]

I am wondering the following question. Let $A$ be a ring. Is it always true that End$_A(A)\cong A$. Is true when $A$ is semisimple? I tried to give an isomorphism $\phi$ between End$_A(A)$ and $A$...
3
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2answers
145 views

$R_R$ is semisimple $\implies$ $M_R$ is semisimple

For any ring $R$ if the right regular $R$-module $R_R$ is semisimple then all right $R$-modules are semisimple. Let $M$ be an artirary right $R$-module. In order to show that $M$ is semisimple I have ...
6
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3answers
444 views

Is every finite dimensional semisimple algebra over $k$ isomorphic to a direct sum of finitely many matrix algebras over $k$?

Let $k$ be a field , let $R$ be a finite dimensional semisimple algebra over $k$ ; is it true that $\exists t \in \mathbb N$ and $n_1,...,n_t \in \mathbb N$ such that $R$ is isomorphic with $\oplus_{...
2
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2answers
77 views

$(eR)_R$ is a semisimple right $R$-module $\implies$ $eRe$ is a semisimple ring

Let $e$ be an idempotent in a semisimple ring $R$. I want to prove that if $(eR)_R$ is a semisimple right $R$-module, then $eRe$ is a semisimple ring. The things I have done so far: Since $(eR)_R$ ...
1
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1answer
28 views

Is $\mathrm{End}_{KS_r}(V)$ semisimple?

If $K$ is a field of characteristic $0$, then the ring $KS_r$ is semisimple. Let $V$ be a $KS_r$-module. In particular it is semisimple. Can we assure that $\mathrm{End}_{KS_r}(V)$ is a semisimple ...
2
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1answer
36 views

(Ex 3.13 Lam) Suppose R is a simple, infinite dimensional algebra over a field k. If V is left R module, then V as vsp over k is infinite dimensional.

My approach: Since R is infinite dimensional over k, therefore the infinite basis are $(r_1,...r_i..)$. Let $r_i.v = w_i\in V$ (Since $V$ is a left $R$ module). Now I prove that $w_i$ are independent. ...
3
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1answer
77 views

(Ex 3.9 b, Lam) If $R$ is semisimple ring with $aR = I$ a two-side ideal, then prove that $Ra = I$.

My approach: Suppose I know that $aR = R$; $a \in R$, $\implies Ra = R$ for a semisimple ring $R$ Now I think I can apply this result as follows. Since $I$ is an ideal in $R$, it can be considered as ...