Questions tagged [semi-simple-rings]
186 questions
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why is a simple ring not semisimple?
A simple module is a semisimple module .
A module $M$ is called semisimple if every submodule is a direct summand of $M$
Since a simple module has $\{0\}$ and $M$ as its submodules so it is semisimple.
But why is a simple ring not semisimple…
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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 an element of $E$.
Here $F$ is a field, and $E/F$…
josephz
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Why $R$ is semisimple ring iff every $R$-module is semisimple?
I'm reading An introduction to homological algebra of Rotman, but the proposition 4.5 of the section 4.1 Semisimple rings states this:
The following conditions on a ring $R$ are equivalent.
$R$ is semisimple.
Every left (or right) $R$-module $M$…
iam_agf
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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 $\oplus_{i=1}^t M(n_i,k)$ as a $k$- algebra ? If not…
user
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7
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A simple ring which is not semisimple
Let $V$ be an $\mathbb{F}$ - vector space with a countably infinite basis. Let $R=\text{End}_R V$ the ring of all linear functions $\phi:V\to V$ and $I=\{f\in R:\, \text{dim}\, f<\infty\}$ the two sided ideal of $R$ consisting of the linear…
ChrisNick92
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$R$ is semisimple iff it is Artinian and $J(R) = 0$
Let $R$ be a ring with identity. The ring $R$ is semisimple if it is semisimple as a left $R$ module. A module $M$ is semisimple if it can be expressed as a direct sum of simple submodules.
The Jacobson radical of $R$, denoted by $J(R)$, is the…
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On semisimple rings
Let $R$ denote a ring with unity.
I know that, if $R$ is semisimple, then every $R$-module is semisimple. In particular the class of indecomposable $R$-modules coincides with the class of simple $R$-modules (If $N$ is indecomposable and semisimple,…
Crostul
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Rings such that every module is a direct sum of generator modules
Is there a classification of those rings $R$ for which the category of left $R$-modules $\mathbf{Mod}(R)$ is generated by a small set of left $R$-modules under direct sums?
For example, every semisimple ring has this property, since every left…
Martin Brandenburg
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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 Maschke's theorem, the group algebra $\Bbb C G$ is…
pancini
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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 $M_n(D)$ is a simple ring for every $n \in\mathbb…
user387219
5
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2 answers
How Zorn's lemma is used here?
I am studying the following theorem in Advanced modern algebra/ Joseph J. Rotman. - Third edition,(Graduate studies in mathematics ; volume 165),
A left $R$ module $M$ over a ring $R$ is semisimple if and only if every submodule of $M$ is a direct…
Infinity_hunter
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5
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1 answer
Module $k[x]/(x-a)^2$ is not semisimple, elegant proof?
Let $k$ be a field and $k[x]$ polynomial ring, and take the module $k[x]/(x-a)^2$ for arbitrary $a\in k$. How to show that this module is not semisimple?
I was thinking the easiest way is to use this (notation from Lang, XVII. Chapter 2. Conditions…
toxic
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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 $\operatorname{ann} M$, the annihilator of $M$, is a maximal left…
StefanH
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5
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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)?, or, What ways to know that a submodule of a…
5
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2 answers
The exact relationship between Clifford algebras and certain special isomorphic k-algebras
question(s):
Choose any real or complex clifford algebra $\mathcal{Cl}_{p,q}$. It's known that there is some $A \simeq \mathcal{Cl}_{p,q}$, where $A$ is either a matrix ring $M(n,R)$ or a direct sum of matrix rings $M(n,R)\oplus M(n,R)$, for $n…
micahscopes
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