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

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1answer
27 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
7
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1answer
61 views

A central division algebra is not its commutator

In looking at old qualifying exam questions, I've come upon a question that has me stumped. Let $A$ be a central division algebra (of finite dimension) over a field $k$. Let $[A,A]$ be the ...
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1answer
33 views

Existence of a maximal submodule

Let $R$ be a left Artinian ring and let $M$ be a nonzero left $R$-module. Prove that $M$ has at least one maximal submodule. I don't really know where to start. Any hint? Thanks!
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2answers
22 views

A direct limit concerning some homomorphisms

In an algebra text there is the following argument I am stuck in the last part of which: "Let $f:B→C$ be an epimorphism in the category of $R$-modules, and $D=∑_{n=1}^∞c_nR$ be a countably generated ...
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1answer
42 views

Isomorphism between $R$ and its dual space

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!
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1answer
48 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
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1answer
42 views

Another description of injective hull

Let $I$ be an injective module containing a module $M$, let $M_1$ be a submodule of $I$ maximal with respect to the property that $M_1∩M=0$, and let $M_2$ be a submodule of $I$ containing $M$ maximal ...
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1answer
58 views

Showing the Weyl algebra is simple.

Let $R$ be a ring with $1$, which contains $\mathbb{Q}$, and generated over $\mathbb{Q}$ by two elements $x,y$ such that $yx-xy=1$. Show that $R$ is simple. What i did? Certainly $x, y \in R$ as ...
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238 views

Why is the Weyl Algebra simple?

Let $\mathbb K$ be a field and $L=\mathbb K\langle x, \frac{d}{dx}\rangle$ be the Weyl algebra over the field $\mathbb K$. That is, the algebra over $\mathbb K$ with two generators $x$ and ...
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2answers
262 views

Isomorphism of an endomorphism ring, how can $R\cong R^2$?

I'm aware that over a commutative ring, two free finitely generated modules of finite rank are isomorphic if and only if they have the same rank. I'm trying to understand a curious example of why ...
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0answers
25 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
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2answers
84 views

One-sided version of the Nakayama lemma?

The Nakayama lemma is often used to show that finitely generated idempotent ideals are generated by an idempotent. What remains true if we go to non-commutative rings? In other words, given a unital ...
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3answers
61 views

Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...
2
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1answer
36 views

Example of a regular element in noncommutative rings

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit?
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0answers
26 views

Example of a regular element with a commutative quotient

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit and $A/(x)$ is commutative?
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1answer
46 views

An injective-injective module problem

I want to prove this problem: For an $R$-module $M$ and an ideal $J⊆R$, let $A=\{m∈M∶mJ=0\}$. If $M$ is an injective $R$-module, show that $A$ is an injective $R/J$-module. We can view $A$ as an ...
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2answers
74 views

Prove that the group $(A,+, ◦) $ is a non-commutative ring

• $A × A → A, (f, g) → f + g$, where $(f + g)(x) = f(x) + g(x)$ for all $x ∈ K$ • $A × A → A, (f, g) → f ◦ g$ where $(f ◦ g)(x) = f(g(x))$ for all $x ∈ K$ Show that $(A,+,◦)$ is a non commutative ...
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1answer
51 views

A condition of equivalence of flatness and projectiveness

This is a problem in "Foundations of Module and Ring Theory" of Wisbauer: " Let $R$ be a subring of the ring $S$ containing the unit of $S$. Show that a flat $R$-module $N$ is projective if and only ...
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1answer
57 views

Why is there no $[X,[X,[X,Y]]]$ and $[Y,[Y,[X,Y]]]$ in the fourth order term of BCH formula?

While trying to deal with a problem involving BCH (Baker-Campbell-Hausdorff) formula, I've noticed something strange. Everywhere in the literature I've managed to fetch (for example: this and this ...
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1answer
52 views

A Direct Sum of Members of a Certain Class of Modules

Let $S$ be a class of $R$-modules and let an $R$-module $M$ be countably generated. Suppose that, for every direct summand $K$ of $M$, each element of $K$ belongs to a direct summand of $K$ that is ...
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3answers
49 views

Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
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1answer
42 views

Show that $D \cong {\rm End}_A(D^n)$ where $D$ is a division algebra and $A\cong M_n(D)$

Define $A$ to be a finite dimensional simple algebra over a field $k$. $D$ is a $k$-division algebra (not necessarily commutative) such that $A\cong M_n(D)$ for some integer $n$. Let $L$ be a minimal ...
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0answers
61 views

BCH (Baker-Campbell-Hausdorff) formula for $[X,Y]=xY-yX$

If some $X$ and $Y$ satisfy the commutation relation $[X,Y]=XY-YX=xY-yX$, where $x$ and $y$ are numbers (or commute mutually and with $X$ and $Y$), then what is the closed form of $\ln(\exp X \exp ...
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2answers
41 views

A doubt about lower nil radical while proving 2-primality of ring.( Baer-McCoy Radical)

I was proving that a reversible ring is 2-Primal for an exercise in T.Y Lam's book, but I got stuck. Here is where I'm stuck: let $a$ be a nilpotent element of $R$ with $a^n=0$. Then using ...
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0answers
42 views

Simple Cogenerator for the category of left $R$-modules

I want to prove that if the category of left $R$-modules has a simple cogenerator $M$, then $R$ is a simple ring. (Here, $R$ is an arbitrary ring with $1$.) My try is to take an ideal $I$ of $R$ ...
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1answer
38 views

if $\frak{u}$ is a left ideal in a simple ring $R$, then $\frak{u}$$\cdot R =R$?

I am reading $\textit{noncommutative rings }$ by Lam. In his proof of Wedderburn-Artin theorem($\S 1.3.11$) he seems to use the following: if $\frak{u}$ is a left ideal in a simple ring $R$, then ...
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1answer
27 views

If $Rx$ is nil then $Rxr$ is nil for any $r \in R$

i am studying kothe's conjecture, ad got stuck here. if $R$ is any non commutative ring, then how is it true that if the ideal $Rx$ is nil then $Rxr$ is nil for any $r \in R$. let $sx\in Rx$, then ...
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1answer
105 views

$z\in\mathfrak R$ iff for every $a\in A$ there is $w$ for which $z+w=zaw=waz$.

In his BAII, Jacobson gives the following exercise, which he attributes to McCrimmon. Show that $z\in\mathfrak R(A)$ iff for each $a\in A$ there exist $w\in A$ such that $z+w=zaw=waz$. I have ...
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1answer
56 views

Identities involving adjoint action

I'm looking for list of identities involving adjoint action $\mathrm{ad}_A X = [A,X] = AX - XA$. For example, it can be easily shown that: \begin{equation} e^{\mathrm{ad}_A} X = e^A X e^{-A} ...
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0answers
32 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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0answers
51 views

Global dimension of translation algebra

What is the Hochschild cohomological dimension of the "translation algebra": $\mathbb{C}\langle x,y\rangle/(xy-yx-x)$? I expect it to be $2$, but I haven;t found a serious argument as to why this ...
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0answers
70 views

Direct product of direct sum of a flat module

I have a problem concerning flat modules: Let $M$ be an $R$-module such that the direct product $M^A$ is flat for all sets $A$. I want to prove that $(M^{(B)})^A$ is also flat for any sets $A$ ...
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1answer
45 views

idempotents acting as local identities

Let $R$ be a ring with unity (not necessarily commutative) and $I$ an ideal of $R$. Suppose that for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is ...
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0answers
53 views

When is an $R$-projective module a projective module?

Let $R$ be a semiperfect ring. Is it a true fact that every $R$-projective module $M$ with $Rad(M)$ superfluous in $M$ is projective? I could not reach a good result using just the fact that ...
2
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1answer
55 views

The reals as an algebra over the rationals

R, the real numbers, is an infinite dimensional commutative division algebra over the rationals Q. Is there an example of an infinite dimensional noncommutative division algebra over the rationals Q?
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1answer
59 views

Locally unital ideals [duplicate]

Let $R$ be a ring with unity not necessarily commutative and $I$ an ideal of $R$. Let for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. ...
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0answers
35 views

Ordered rings $R$ with $ab = ba$, $ab \leq ba$ or $ba \leq ab$ for all $a, b \in R$?

Ordered rings $R$ with $ab = ba$, $ab \leq ba$ or $ba \leq ab$ for all $a, b \in R$? Are there any good examples that are not also commutative rings? I can't seem to think of any.
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0answers
35 views

Graduations and filtrations for localizations

I'm trying to answer the following questions: Let $A$ be a (not necessarily commutative) $\mathbb{Z}$-graded ring and $S$ a multiplicative subset of $A$ such that $AS^{-1}$ exists. Is $AS^{-1}$ a ...
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0answers
40 views

Ideals of infinite product rings

(Rewritten following earlier feedback...) (1) What are the ideals of ${\mathbb Z}^{\mathbb N}$? That is, take the ring which is the product of a countable infinity of copies of $\mathbb Z$; is there ...
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1answer
23 views

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.
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1answer
69 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
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1answer
96 views

Does an ideal of finite codimension in a finitely generated algebra have always to be finitely generated?

I have been reading a book on Lie Algebras ("Álgebras de Lie" by San Martin) and there is this exercise in the chapter on universal enveloping algebras with a claim that I can not prove: Suppose ...
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2answers
51 views

Building quotient rings

The quotient rings are following: $\mathbb{Z}[i]/(1+i)$, $\mathbb{Z}[i]/(1+2i)$, $\mathbb{Z}[\sqrt{-2}]/(2)$, $\mathbb{Z}[\sqrt{-2}]/(1+ \sqrt{-2})$. I know that the two first are likely to be ...
2
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1answer
27 views

Simplicity is invariant under extension of scalars

Problem Suppose $A$ is a central simple $k$-algebra, which means that the field $k=Z(A)$ and $A$ is itself a simple ring, where $Z(A)$ is the center of $A$, and $K/k$ is a field extension, then ...
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1answer
40 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
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1answer
43 views

Localization of a direct product

Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
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13 views

Finitely generated left (right) unitary modules over left(right)-Artinian ring with identity are Artinian [duplicate]

Finitely generated left (right) unitary modules over left(right)-Artinian ring $R$ with identity are Artinian. How to prove it?
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1answer
34 views

Associative ring with identity, inverses, divisors of zero and Artinianity

How to prove the following? $R$ is an associative ring with identity. $R$ contains element $r$. The element is not invertible on the right and is not a left divisor of zero. Then the ring $R$ cannot ...
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1answer
26 views

Commutativity of comodules

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\otimes_R B \cong B \otimes_R A$ as $R-modules$. However, this also true ...
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2answers
56 views

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 ...