2
votes
3answers
39 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
36 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
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 ...
4
votes
1answer
53 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
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 ...
1
vote
1answer
44 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
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.
-2
votes
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. ...
3
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
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 ...
5
votes
1answer
62 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
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 ...
0
votes
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 ...
0
votes
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?
2
votes
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 ...
0
votes
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.
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
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
78 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
76 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
61 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
39 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
46 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
46 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
28 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
91 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
100 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 ...
2
votes
1answer
50 views

If $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism, which cases is true?

Let $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism,then which cases is true? $S$ is left Artinian $S$ is left Noetherian $S$ is simple ring ...
2
votes
1answer
62 views

Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring. Now suppose that I have a non-commutative ring $N$. Suppose ...
0
votes
1answer
90 views

Relation between Jacobson radical and composition series

Let $R$ be a not necessarily commutative ring with 1. Suppose $R$, viewed as a right $R$-module, has a finite composition series with non-isomorphic composition factors. Prove that the Jacobson ...
10
votes
2answers
112 views

Real forms of complex vector spaces and $\mathbb{C}$-algebra

A real form $W$ of a complex vector space $V$ is a real subspace s.t. $\mathbb{C}\otimes_{\mathbb{R}}W \cong V$ by $a\otimes x \longrightarrow ax$, or equivalently there is an $\mathbb{R}$-basis of ...
1
vote
0answers
43 views

Enveloping Algebra equal to algebra

Let $R$ be a unital associative ring, $A$ be an associative $R$-algebra of finite dimension, and $A^e$ its enveloping algebra. What are the requirements on $A$, so that $A^e \cong A$ (as ...
0
votes
1answer
69 views

What is needed to force polynomial-ring automorphisms to be affine?

Is there an integral domain $R$ and a polynomial-ring automorphism $\: \phi : R[x] \to R[x] \:$ such that, for $\: i : R\to R[x] \:$ the canonical embedding, $\;\;\; \phi \circ i \: = \: i \;\;$ and ...
0
votes
0answers
100 views

Rings, annihilators and (maximal) ideals

Let $R$ be a unital, associative, non-commutative ring. If $P$ is an ideal of $R$, what is the annihilator of quotients $R/PR$ and of $R/P$? Does something change if $P$ is supposed to be a maximal ...
5
votes
2answers
92 views

if $A^\times $ is a commutative group, does $A$ necessarily be a commutative ring?

Let $A$ be a ring and $A^\times$ be the collection of unit elements of $A$. If $A$ is a commutative ring, then $A^\times$ is a commutative group. Conversely, if $A^\times $ is a commutative group, ...
2
votes
1answer
91 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
1
vote
1answer
59 views

Clarification about some proof of Projectivity

Small provides an example of a ring which is right but not left hereditary is the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q} \end{matrix} \right)$; To ...
7
votes
0answers
146 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...
0
votes
0answers
38 views

Basic facts about finitely-generated noncommutative algebras

I don't really have any feeling for non-commutative rings, so I just wanted to check that I'm sane. A simple "yes" or "no" will probably do, unless I'm doing something incredibly stupid in which case ...
15
votes
4answers
617 views

Smallest non-commutative ring with unity

Find the smallest non-commutative ring with unity. (By smallest it means it has the least cardinal.) I tried rings of size 4 and I found no such ring.
6
votes
1answer
76 views

A finite unital and commutative ring with exactly one maximal ideal has $p^{n}$ elements.

Suppose $R$ is a finite unital and commutative ring that has exactly one maximal ideal. Prove that $\left | R \right |=p^{n}$ where $p$ is a prime number. If $R$ will be non-commutative, do we have ...
6
votes
1answer
80 views

P(R) is contained in Nil(R) for noncommutative rings.

How to show that $P(R)$ is contained in $\operatorname{Nil}(R)$ (where $R$ is a noncommutative ring with identity)? Definitions I am using: A nil right ideal is one whose elements are all ...
0
votes
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 ...
2
votes
3answers
99 views

Example of a non commutative domain that is not a division ring

My experience with non commutative rings is limited to 2 by 2 matrices and the quaternions. The first of which is not a domain, and the latter is a division ring. I'm looking for an example of a ...
1
vote
0answers
36 views

Every onesided nilideal of a right noetherian ring is nilpotent.

Suppose $R$ is a right noetherian ring. Prove that every onesided nilideal is nilpotent. I try to use this theorem: If R is a commutative Ring and I is nilideal of R and also I is finitely ...
2
votes
1answer
87 views

Is every regular element of a ring invertible?

When I am reading a paper, I found the definition of a new ring as following: In this definition, if every central regular element is invertible, i.e., how to understand the invertible element of u? ...