1
vote
1answer
32 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
25 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
36 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
85 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
49 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
45 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
77 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
105 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
41 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
63 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
80 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
81 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, ...
1
vote
1answer
65 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
56 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 ...
6
votes
0answers
137 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
26 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 ...
14
votes
4answers
497 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
71 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
79 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 ...
2
votes
3answers
90 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
27 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
83 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? ...
2
votes
1answer
61 views

Prove that $Re+Rf=Re\oplus R(f-fe)$

Let $e,f$ be idempotent elements of a ring $R$. Prove that $Re+Rf=Re\oplus R(f-fe)$. This post solves the first part of my question. How should we prove the second part, that $e,f$ are ...
0
votes
1answer
60 views

Non-commutative indeterminates in polynomial rings.

Described below are some observations I have made while fiddling around with polynomials. In addition to the two questions below, I am looking for any sort of relevant information so I can read more. ...
0
votes
0answers
59 views

Show that Weyl algebra is noetherian

Let $k$ be a field. I want to show that the ring $D=k\left[x_1,x_2,\dots,x_n,\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\dots,\frac{\partial}{\partial x_n}\right]$ which acts on ...
2
votes
1answer
72 views

Problem with semisimple ring theorem

Proposition: For a ring $R$ the following statements are equivalent: (a) $R$ has a simple left generator; (b) $R$ is simple left artinian; (c) For some simple $_RT, _RR \cong ...
3
votes
1answer
89 views

Problem with Wedderburn Theorem proof.

This is a Wedderburn Theorem proof in Frank W. Anderson, Kent R. Fuller: Rings and Categories of Modules Please explain that: "Therefore $_RR$ has a composition series of length $n$". Exercise ...
7
votes
3answers
267 views

Is it possible that $(ab)^{-1}$ is defined although $a^{-1},b^{-1}$ are not?

I wish to enquire about the properties of units in abstract algebra. In a ring $R$, a unit $u$ is an invertible element. Let $u=ab$. Is it possible that $a$ and $b$ are not units? Is it possible ...
1
vote
1answer
33 views

Existence of Algebra of anticommuting idempotents

Background and motivation: I'm wondering about the existence of an algebra which is in some ways similar to the exterior algebra, but is generated by idempotents rather than nilpotents. Let $V$ be a ...
5
votes
2answers
132 views

Centre of a simple algebra is a field

How can one show that the centre of simple algebra is a field? I have tried it and proved that the inverse exists for every element of centre but cannot prove that inverse of every element ...
1
vote
2answers
99 views

Skew Laurent Polynomial Ring.

Let $R$ be a ring and $R[x^{\pm 1}]$ the Laurent Polynomial Ring. $R[x^{\pm 1}]$ is a domain since $R$ is. How to show this? Let $R$ be a ring and $R[x^{\pm 1}]$ the Laurent Polynomial Ring. If ...
20
votes
1answer
637 views

A ring with few invertible elements

Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field.
3
votes
1answer
111 views

Left and Right Ideal Generated by Two Matrices.

Let $R= {\rm Mat}_2(\Bbb R)$ be the ring (with $1$) of $2\times2$-matrices with entries in $\Bbb R$. Let $$M = \left\{\begin{pmatrix}1&0 ...
1
vote
0answers
35 views

For certain prime ideal $\wp$, $(\wp^e)^c=\wp$?

Let $R\subset S$ be noncommutative rings, and let $\wp$ be a prime ideal in $R$, denote $\wp^e$ to be the extension ideal of $\wp$ in $S$, then my question is: Does $(\wp^e)^c:=\wp^e\cap R=\wp$ hold ...
14
votes
1answer
249 views

Have I found a counterexample to Noether-Skolem? (No, but I am confused…)

I was toying around with central simple algebras over a field $K$ today and thought that I should try to verify Noether-Skolem's theorem that any automorphism of such must be inner. So, let us take $K ...
1
vote
1answer
39 views

Simple components of a semisimple $K$-algebra

Let $A$ be a finite dimensional semisimple $K$-algebra. Since $A$ is semisimple, I can write it as a direct sum of its simple components: $$A=A_1\oplus\cdots\oplus A_m$$ where each $A_i$ is of the ...
2
votes
1answer
51 views

Is there a right semihereditary domain which isn't right Ore?

I do not have a lot of examples in my head for semihereditary domains at all, and I haven't been able to see how to resolve this question: Is there a right semihereditary domain which isn't right ...
2
votes
2answers
239 views

What does Herstein mean by 'centroid of a ring'?

I'm currently reading Herstein's Noncommutative Rings, and the definition of the centroid of a ring is on page 46 of the book. Let $\text{End}(R)$ be the ring of endomorphisms of the additive group ...
5
votes
1answer
78 views

Simple + Artinian = Semiprimitive

By a noncommutative ring I mean that it has no unit. I know that if some ring (say, $R$) is simple, then: $R^2 \neq (0)$ It only possesses $2$ two-sided ideals, namely $(0)$, and itself. And ...
6
votes
3answers
439 views

Left Multiplication Ring Homomorphism

Assume we have a non-commutative unital ring $R$ and an element $r$ not in the center. Define a map $$\phi_r:R\rightarrow R$$ $$x\mapsto rx$$ Can this ever be a ring homomorphism? If it can be ...
4
votes
2answers
75 views

$I$ semisimple + $R/I$ semisimple $\implies$ $R$ semisimple

Let $R$ be a (not necessarily commutative) ring with unit. Let $I\subset R$ be an ideal that in turn is a ring with unit. Is there a theorem that says something like $I$ semisimple and and $R/I$ ...
-2
votes
1answer
76 views

An example of Von Neumann regular ring with nilpotent element in noncommuative ring.

Give an example of Von Neumann regular ring with nilpotent element.
3
votes
2answers
98 views

When a group algebra (semigroup algebra) is an Artinian algebra?

When a group algebra (semigroup algebra) is an Artinian algebra? We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group ...
16
votes
1answer
187 views

How 'commutative' can a non-commutative ring be?

Let $R$ be a finite non-commutative ring. Let $P(R)$ be the probability that two elements chosen uniformly at random commute with each other. Consider the value $$S=\sup_RP(R)$$ where the supremum ...
16
votes
2answers
298 views

Example of unital non-commutative ring with $(ab)^2=(ba)^2$ for all $a,b$

I'm trying to exhibit a unital, non-commutative ring $R$ such that $(ab)^2=(ba)^2$ for all $a,b\in R$. This is an exercise out of Herstein's Topics in Algebra. In the previous exercise, I showed ...
2
votes
2answers
118 views

Multiplicity of the simple $R$-module $M$ in the semisimple ring $R$

I'm confused about the conclusion of Wedderburn's structure theorem for semisimple rings. Let's consider the special case where $R=M^n$ as modules for some simple module $M$. Wedderburn's theorem says ...
0
votes
0answers
216 views

Are the elements of a division algebra which commute with all commutators in the center of the algebra?

Let $G$ be a division algebra. If an element $c$ of $G$ commutes with every commutator of $G$, then $c$ is in the center of $G$. Is it true? If it is true, how to prove? Here commutator means an ...
1
vote
2answers
102 views

tensor product and direct product of algebra presentations

Let $R$ be a commutative unital ring and $R\langle x_i\mid f_j\rangle$ denote a unital $R$-algebra presentation. Q1: What is the presentation of $R\langle x_i\mid f_k\rangle\otimes R\langle y_j\mid ...
2
votes
1answer
43 views

In the lattice of ideals, what are the lowerbounds of the prime ideals?

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the ...
5
votes
2answers
220 views

Prove that semi-simple rings are Dedekind-finite

Just to be consistent with the terminology, let me define the words I'm using. A module $M$ is simple if it has no proper non-zero submodules, and semi-simple if it can be written as a direct sum $M ...