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

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1answer
37 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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3answers
34 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, ...
2
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1answer
39 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
41 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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1answer
48 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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0answers
28 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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0answers
35 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
35 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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2answers
36 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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1answer
26 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
52 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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1answer
50 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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0answers
31 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
48 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
66 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
43 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
49 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
53 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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0answers
34 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
23 views

locally unital ideals of ringa [duplicate]

1 down vote favorite Let R be a ring with unity not necessarily commutative and I an ideal of R.Let for every element a of I there exists an element c of I such that ac=a.Note that c is related to a. ...
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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

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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1answer
59 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
20 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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2answers
46 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 ...
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1answer
39 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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0answers
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
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
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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1answer
33 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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2answers
38 views

unique maximal semisimple submodule

I read a property An $R$-module $M$ has a unique maximal semisimple submodule. I am not sure whether R as a ring needs to be commutative or not. How to proof it?
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1answer
16 views

a ring which is right primitive but not left primitive.

i am doing the example constructed by BERGMAN in 1964 (see below for link), and i have a little doubt , he defines r.s=rs in Q(X) but Q(X) must have addition as operation as it is being checked for ...
2
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1answer
94 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 ...
1
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1answer
33 views

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 ...
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1answer
14 views

doubt in example of left primitive ring.

According to Lam, let k be any division ring, V be a right k-vector space, and E=End(V), operating on the left of V. then it says clearly V is faithful simple left E module, so E is left primitive ...
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1answer
46 views

Is there a (hypercomplex) number system, in which addition is **not** commutative

Now, we all know and love the quaternions, in which multiplication is not commutative, and the fact that, in all Clifford algebras, exponentiation is not commutative. Having looked at the properties ...
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1answer
37 views

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 ...
3
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1answer
135 views

Pure states and irreducible *-representations

If $A$ is a $C^*$-algebra, then a $*$-representations of $A$ is irreducible if and only if the corresponding state is a pure state. What happens if don't insist on $A$ being a $C^*$-algebra? Is one ...
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0answers
28 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
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1answer
34 views

a doubt in wedderburn artin theorem

i am doing proof of wedderburn artin theorem from T Y Lam but the fact used in proof is decomposing semisimple Ring R as FINITE direct sum of minimal left ideals, but in in definition it is said to be ...
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2answers
31 views

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 ...
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1answer
48 views

Is this algebra a simple ring?

Is the algebra $ \mathcal{M}_{3} ( \mathbb{C} [X,Y,Z] ) $ a simple ring (a simple algebra)? $ \mathcal{M}_{3} ( \mathbb{C} [X,Y,Z] ) $ is the matrix algebra over $ \mathbb{C} [X,Y,Z] $. Thanks a ...
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0answers
37 views

Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
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0answers
33 views

Simple Modules over the Weyl Algebra

Let $k$ be a field of characteristic zero and let $A_1=k\langle x,y| \, xy-yx=1 \rangle$ be the Weyl algebra. Is there a (more or less explicit) possibility of writing down all simple modules over ...
5
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1answer
92 views

Which mathematical structures are particular cases of small categories?

In what follows, all categories are assumed to be small (classes of objects and morphisms are sets). Which mathematical structures $X$ can be seen as particular cases of small categories ...
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1answer
42 views

Annihilator of a quotient module

Let $J$ be an ideal of $R$, and $M$ a right $R$-module. Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine ...
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1answer
36 views

Question about an argument in Isaac's Algebra: Wedderburn-Artin Theorem

Theorem: Let $R$ be a Wedderburn ring, and let $S$ be a representative set of the simple right $R-$modules. For each $S \in S$, let $I_S$ be the sum of all those minimal right ideals of $R$ that are ...
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0answers
27 views

Any counterexample for inverse limit functor not to be right exact [duplicate]

We know that inverse limit is a "left" exact functor on the category of modules in the sense that whenever $r:(A_i,α_j^i )→(B_i,β_j^i )$ and $s:(B_i,β_j^i )→(C_i,γ_j^i )$ are transformations of ...
1
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1answer
25 views

Is there a direct proof that $M_n(\mathcal k)$ is semisimple ring.

An R-module M is called semisimple if on of the following condition holds: 1) M is a direct sum of simple* submodules of M 2) M is a sum of simple submodules of M 3) For any R-submdoule N of M ...
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2answers
55 views

Injective resolution for an integral domain

How could one write an injective resolution for an arbitrary commutative integral domain $R$? Thanks in advance!