1
vote
1answer
25 views

$R/J(R)$ not semisimple Artinian

I search for a ring $R$ with Jacobson radical $J(R)$ such that $R/J(R)$ is not semisimple Artinian. Being a finitely generated module over itself, $R$ would have infinite hollow dimension due to ...
1
vote
1answer
57 views

Give an example of a maximal ideal in a noncommutative ring which is not prime

While trying to find an example, I came up with this: Since if $J$ is an ideal of a the ring $M_n(R)$, where $R$ is a commutative ring, then $J=M_n(I)$ for some ideal $I$ of $R$. IF I could show that ...
2
votes
1answer
49 views

Zerodivisors in polynomial rings over a non-commutative ring

Prove that if $f \in R[x]$ is a zero divisor then $\exists r(\neq 0) \in R$ s.t $rf=0$, where $R$ is a ring. I know that for $(a_0+a_1x+ \cdots ...
2
votes
1answer
29 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
0
votes
0answers
41 views

Identifying a Lie algebra from its universal enveloping algebra

Its been a while since I've worked on Lie algebras and I can't remember how to approach this problem: How do I identify the lie algebra (up to isomorphism) associated to a certain universal ...
0
votes
1answer
30 views

Show that the probability that two randomly choosen elements from $R$ commute is at most $(5/8)$.

Let $R$ be a finite non-commutative ring. Show that the probability that two randomly choosen elements from $R$ commute is at most $(5/8)$. i know there is an answer in ...
1
vote
1answer
36 views

Jacobson radical of a matrix ring

I search for a way to prove that the Jacobson radical of $R=\left [\begin{array}\ \mathbb Z_4 & 2\mathbb Z_4 \\ 0 & \mathbb Z_4 \end{array} \right ]$ is $\left [\begin{array}\ 2\mathbb Z_4 ...
0
votes
0answers
34 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
13
votes
1answer
194 views

Rings with $a^5=a$ are commutative

Let $R$ be a ring such that $a^5=a$ for all $a \in R$. Then it follows that $R$ is commutative. This is part of a more general well-known theorem by Jacobson for arbitrary exponents ($a^n=a$), which ...
1
vote
0answers
21 views

Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
3
votes
2answers
83 views

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
0
votes
0answers
46 views

An error in the book “noncommutative ring” writed by Herstein

I'm reading the book "noncommutative ring" writed by Herstein. In the page 15, the author says that Let $F$ be a field and $A$ is an algebra over $F$. Let $\rho$ be a maximal regular right ideal ...
7
votes
1answer
75 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 ...
2
votes
1answer
49 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.
2
votes
1answer
63 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 ...
1
vote
0answers
26 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 ...
2
votes
3answers
64 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 ...
7
votes
2answers
87 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 ...
1
vote
0answers
28 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?
2
votes
1answer
39 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?
2
votes
3answers
51 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, ...
1
vote
0answers
46 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 $R$ ...
3
votes
2answers
48 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 ...
4
votes
1answer
143 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 ...
2
votes
0answers
34 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 ...
1
vote
1answer
46 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 ...
2
votes
0answers
38 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.
2
votes
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 $a$. ...
1
vote
0answers
36 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 ...
5
votes
2answers
82 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 ...
1
vote
2answers
52 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 ...
0
votes
1answer
44 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 ...
0
votes
0answers
14 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?
2
votes
1answer
28 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 ...
0
votes
1answer
50 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.
0
votes
1answer
44 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 ...
2
votes
1answer
59 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 ...
1
vote
0answers
29 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 ...
0
votes
1answer
53 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 ...
1
vote
1answer
29 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 ...
1
vote
1answer
55 views

A statement equivalent to flatness

If $R$ is a ring with identity and $P$ is a flat right $R$-module, it is a fact that any $R$-homomorphism $f$ from a finitely presented right $R$-module $M$ to $P$ factors through a finitely generated ...
1
vote
2answers
104 views

Is the center of a ring an ideal?

Let $Z(R) = \{ a \in R : ax = xa,\text{ for all $x \in R$}\}$ Is $Z(R)$ an ideal of $R$? Attempt: I already proved that $Z(R)$ is a subring of $R$. I would say yes, since if $x \in R$, then $xa$ is ...
2
votes
1answer
91 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
3
votes
1answer
64 views

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's $Algebra$ ...
2
votes
1answer
50 views

Determine up to isomorphism all semisimple noncommutative rings with order 512

Problem: Determine up to isomorphism all semisimple noncommutative rings of order 512 = $2^9$. (This is problem from an old qualifying exam I am studying from) So far I have: Let A be a semisimple ...
1
vote
1answer
51 views

Noncommutative finitely generated algebras need not be noetherian

I would like to understand an example (of the title) given in the book "An Introduction to Noncommutative Noetherian Rings" by K. R. Goodearl, R. B. Warfield... On page 8, Exercise 1E, an example of ...
1
vote
1answer
55 views

Example of excision in Hochschild homology

The excision theorem for Hochschild homology introduced by Wodzicki seems like a very powerful tool (as scision was hyper-useful in topology). However, I cannot actually seem to think of a result ...
1
vote
0answers
31 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
1
vote
1answer
144 views

Is every left maximal ideal the annihilator of a simple left module?

In my version of Noncommutative Algebra, by Benson Farb & R. Keith Dennis, in chapter I, section 2 on the Jacobson radical, it is claimed that … each maximal left ideal $I$ is the annihilator ...
1
vote
0answers
105 views

Relation between finite stable rank and IBN (invariant basis number)

For R is any ring has an identity, we known if R has stable rank one, then R is "weakly finite" (or "stably finite," all matrix rings over R are Dedekind finite) and this implies R has IBN . But ...