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Questions tagged [noncommutative-algebra]

For questions about rings which are not necessarily commutative and modules over such rings.

In abstract algebra, the theory of rings which are not necessarily commutative is called "noncommutative algebra." In this way it is a generalization of commutative algebra. Some results from commutative algebra hold in noncommutative algebra, but many results break down.

The ring of quaternions was among the first motivating examples of noncommutative rings. Other familiar examples include the $n\times n$ matrix ring over any ring ($n>1$).

A few examples of some differences between commutative and noncommutative algebra:

  • If $R$ is a commutative ring, and $R^n\cong R^m$ as $R$ modules for some positive integers $m$ and $n$, then $m=n$. In contrast, there is a noncommutative ring such that $R^m\cong R^n$ for every pair of positive integers $m,n$.

  • Any commutative ring without zero divisors can be embedded in a field. There are examples of noncommutative rings without zero divisors which cannot be embedded into a division ring. This is one of many signs that show localization does not work well for many noncommutative rings.

  • The module $R_R$ may have different properties from the module $_RR$. For one thing, one could be Noetherian (or Artinian) without the other having the same property.

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Universal enveloping algebra of a Poisson algebra

For a Lie algebra, $\mathfrak{g}$, one has an equivalence of categories between Mod($\mathfrak{g}$) and Mod($U(\mathfrak{g})$), where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$. Let $P=(A,\{,\})$ be a Poisson algebra…
J. Gaddis
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Equivalent definitions of the Jacobson Radical

Noncommutative Algebra / Farb & Dennis defines the Jacobson Radical $J(R)$ of a ring $R$ as the intersection of all annihilators of simple $R$-modules (page 58). It is then claimed that $J(R)$ is also the intersection of all maximal left ideals in…
Amy
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Problem 24 of section 2 of *Noncommutative Algebra* by Farb & Dennis

Problem 24 of section 2 of Noncommutative Algebra by Farb & Dennis states: Let $R$ be an artinian ring and let $G$ be a finite group. Show that $R[G]$ is semisimple if and only if $R$ is semisimple and $|G|$ is invertible in $R$. I tried to make…
Jen
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Ring with Unique Simple Module

Let $A$ be a not necessarily commutative unital ring with a unique simple module (up to isomorphism). Let $\mathfrak m$ be the annihilator of this simple module, which is a two-sided ideal. We claim that $\mathfrak m$ is a maximal two-sided ideal.…
Nishant
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does simple ring implies artinian ring?

so my doubt is that i am studying wedderburn artin theory and it gives structure of simple artinian rings, but if a ring is simple, it has no nonzero proper 2-sided ideals so it satisfies DCC on ideals trivially, so must be artinian, so if every…
Bhaskar Vashishth
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Finite semisimple rings

I am trying to solve a problem that asks to classify the semisimple rings of order $5^4$. It is a question of a past exam and I am not really sure about the answer. What I know; semisimple rings are finite direct products of matrix algebras over…
user128787
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A simplified definition and examples of prime radical of a non-commutative ring

please provide a definition and some examples of prime radical (or Baer-McCoy radical or lower nilradical) of a non-commutative ring.please be specific.
ujjwal
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When does multiplication with an ideal commute with the product of modules?

I have tried for some time now to prove the following statement from an exercise, and now I wonder if it is even correct: Let $A$ be a ring and $E$ a left $A$-module. For a left ideal $\mathfrak{a}$ of $A$, $\mathfrak{a}E$ is defined as the…
Stefan
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$\mathfrak{u} \cdot R =R$ for every left ideal $\mathfrak{u}$ in a simple ring $R$

I am reading Noncommutative Rings by Lam. In his proof of the Wedderburn-Artin theorem (§1.3.11) he seems to use the following: If $\mathfrak{u}$ is a left ideal in a simple ring $R$, then $\mathfrak{u} \cdot R = R$. I wonder why this is true. I…
Alex
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Skolem-Noether theorem

Skolem-Noether Theorem: Let S be a finite dimensional central simple k-algebra, and let R be a simple k-algebra. If f,g: R-> S are homomorphism (necessarily one-to-one), then there is an inner automorphism T:S -> S such that Tf=g. What happens if we…
Çiğdem
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Proving this quotient of an endomorphism ring is simple.

Let $V$ be a right $D$ (a division ring) vector space of countably infinite dimension. Let $E=\mathrm{End}(V_D)$, then I is the ideal of E consisting of endomorphisms of finite rank. The claim is that $R=E/I$ is a simple ring. So I take an ideal in…
Bhaskar Vashishth
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Noncommutative ring with eight elements and with unity

I'm a bit confused about an exercise I read. Namely, T. Y. Lam's A First Course in Noncommutative Rings has the following on page $23$ Ex. $1.10$. Let $p$ be a fixed prime. Show that there exists a noncommutative ring (with identity) of order…
studying
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Does quotient commute with localization for non-commutative rings?

Following on from the question Does quotient commute with localization?, I'm interested in doing the same sort of thing but over non-commutative rings. Is there a non-commutative analogue of the exact sequences result used in that question? …
Eithil
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Could anyone suggest some examples of $GL_n$-invariant non-trivial two-sided ideals of finitely generated free associative algebras?

Let $k$ be a field and $F=k \langle x_{1},\ldots,x_{n}\rangle $ be a finitely generated free associative algebra. Let $g = (g_{ij})$ be any matrix of the group $GL_n(k)$ and define the algebra automorphism $\rho_g:F\to F$ such that $x_i\mapsto…
kumar
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If $A^n$ has a free subset of $n+1$ elements, it has an infinite free subset.

I want to prove the following: Let $A$ be a ring and $n$ a natural number. If the left $A$-module $A^n$ contains a free subset of $n+1$ elements, then $A^n$ already contains an infinite free subset. Since we can embed $A^{n+1}$ into $A^n$, we can…
Stefan
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