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

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42
votes
6answers
1k views

Is there a non-commutative ring with a trivial automorphism group?

This question is related to this one. In that question, it is stated that nilpotent elements of a non-commutative ring with no non-trivial ring automorphisms form an ideal. Ted asks in the comment for ...
28
votes
1answer
2k views

An example of a division ring $D$ that is **not** isomorphic to its opposite ring

I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
21
votes
1answer
674 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.
19
votes
3answers
303 views

Does my definition of double complex noncommutative numbers make any sense?

I wanted to factorize $a^2+b^2+c^2$ into two factors in a similar way to $$a^2+b^2 = (a+ib)(a-ib)$$ This doesn't seem to be possible using real or complex numbers. However I came up with the following ...
16
votes
4answers
669 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.
16
votes
1answer
210 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
342 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 ...
14
votes
2answers
1k views

(Organic) Chemistry for Mathematicians

Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do ...
14
votes
1answer
259 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 ...
14
votes
1answer
266 views

Ideal in an Artinian Ring $I=aR=Rb$, prove $I=Ra=bR$

Let $R$ be an Artinian Ring and suppose there exists $a,b\in R$ s.t. $I=aR=Rb$, then prove $I=bR=Ra$. (You may assume that a right Artinian Ring is Right Noetherian). I've managed to get $Ra$, ...
12
votes
2answers
578 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
11
votes
1answer
203 views

Does this notion of morphism of noncommutative rings appear in the ring theory literature?

Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
11
votes
1answer
287 views

What are the rings in which left and right zero divisors coincide called?

A unital ring $R$ is reversible iff $ab=0\implies ba=0.$ This condition implies the following one. If $a\in R$ is a left-zero divisor, then $a$ is also a right-zero divisor. And the other way ...
10
votes
4answers
692 views

$AB \neq 0$ but $BA=0$

Do there exists to matrices or objects such that $AB \neq 0$ but $BA=0$? Another way to ask this question is if there exists objects or matrices $A$ and $B$ such that... $[A,B]=AB$ where $[ \, , \, ]$ ...
10
votes
2answers
114 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 ...
9
votes
2answers
518 views

simple ring which is not semisimple

Let $V$ be a vector space of countably infinite dimension over a field $k$ and put $R = \text{End}_k(V)$. Then it is not hard to see that $R$ has a unique proper nonzero two-sided ideal $I$, which ...
9
votes
1answer
389 views

A graded ring $R$ is graded-local iff $R_0$ is a local ring?

Update: I've copied this question over to mathoverflow.net: http://mathoverflow.net/questions/100755/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring to see if I get any answers there. Let ...
9
votes
1answer
240 views

A ring that has exactly 7 left ideals (T. Y. Lam)

Exercise 3.25 in Lam's First Course states: Let $R$ be a ring that has exactly seven nonzero left ideals. Prove that one of them is an ideal (i.e. left and right) and provide an example of such a ...
8
votes
3answers
862 views

Commutative property of ring addition

I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring: Let $R\neq ...
8
votes
2answers
402 views

Coproduct in the category of (noncommutative) associative algebras

For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras ...
8
votes
1answer
283 views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are: How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
7
votes
3answers
288 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 ...
7
votes
1answer
68 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 ...
7
votes
2answers
485 views

Why are Dedekind-finite rings called so?

A Dedekind-finite ring is a ring in which $ab=1$ implies $ba=1$. It seems natural to look for a connection to Dedekind-finite sets, however for such a set any injective endomorphism is surjective, ...
7
votes
1answer
151 views

Proper ideals generated by central ideals

Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly ...
7
votes
2answers
86 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 ...
7
votes
1answer
146 views

Projective objects in the category of rings

What are the projective objects in the category of rings with identity ? Remarks: The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$. If $R$ is projective and ...
7
votes
2answers
318 views

Matrices which commute with all the matrices commuting with a given matrix

Let $A$ be an $n \times n$ matrix with entries from an arbitrary field $F$ and let $C(A)$ denote the set of all matrices which commute with $A$. Is it true that $C(C(A))= \{ \alpha_1 + \alpha_2 A ...
7
votes
0answers
65 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 ...
6
votes
2answers
120 views

How does the lie algebra capture compactness of the lie group?

This is a soft question. Let $V,\rho$ be a representation of the lie algebra $\mathfrak{so}_3(\mathbb{R})$. Then if I understand everything right, $V$ is necessarily completely reducible, because the ...
6
votes
3answers
460 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 ...
6
votes
2answers
111 views

Representing sums of matrix algebras as group rings

Let $A = M_{n_1}(\mathbb R) \oplus M_{n_2}(\mathbb R) \oplus ... \oplus M_{n_m}(\mathbb R)$ be a direct sum of real matrix algebras. Under what conditions does there exist a group ring $\mathbb R[G]$ ...
6
votes
2answers
91 views

What assumptions are needed to prove that $(ab)c=b(ac)$ implies commutativity & associativity?

The question is in the title: I would like to know what extra assumptions (if any) are needed for the following derivation that $(a\cdot b)\cdot c=b\cdot(a\cdot c)$ implies commutativity, i.e. ...
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 ...
6
votes
2answers
248 views

Example of a commutative perfect ring that is not artinian

I read a result here stating that a commutative perfect ring is artinian if and only if it is $(1,1)$-coherent (see Proposition 5.3). I'm interested in finding an example of a commutative perfect ...
6
votes
1answer
79 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
3answers
523 views

The Jacobson radical and quasi-regular elements in polynomial rings — trouble understanding a proof.

The concept of the Jacobson radical is quite new to me and I have to give a talk about a certain paper concerning it. I agreed to do this to find motivation to finally study this concept properly, as ...
6
votes
1answer
539 views

Supplementary exercises for Herstein's Noncommutative Rings

I've been studying from the book Noncommutative Rings by Herstein (not as a part of some official course), but unfortunately it doesn't contain any exercises apart from a few simple ones in the body. ...
6
votes
1answer
94 views

Minimal right ideals of a simple ring are generated by idempotents

Let $M$ be a minimal right ideal of a simple ring $R$. Show that $M$ contains an idempotent $e$ such that $M = eR$.
6
votes
0answers
74 views

Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
6
votes
0answers
154 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 & ...
5
votes
3answers
156 views

Can the product of non-zero ideals in a unital ring be zero?

Let $R$ be a ring with unity and $0\neq I,J\lhd R.$ Can it be that $IJ=0?$ It is possible in rings without unity. Let $A$ be a nontrivial abelian group made a ring by defining a zero multiplication ...
5
votes
2answers
130 views

Commutative/noncommutative algebra?

I know basic knowledge of undergraduate algebra till galois theory of finite extensions. I want to learn number theory, but also like algebra. This semester I have to choose to read either commutative ...
5
votes
3answers
124 views

Is the center of the universal enveloping algebra generated by the center of the lie algebra?

Let $\mathfrak{g}$ be a Lie algebra over a field $k$, and let $U(\mathfrak{g})$ be its universal enveloping algebra. $\mathfrak{g}$ is canonically embedded in $U(\mathfrak{g})$; identify it with its ...
5
votes
2answers
87 views

Showing that $\operatorname {Br}(\Bbb F_q)=0$

I want to prove that $\operatorname {Br}(\Bbb F_q)=0$ using the cohomological description of the Brauer group. We have: $\operatorname {Br}(\Bbb F_q)=H^2(\operatorname {Gal}(\overline {\Bbb ...
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, ...
5
votes
1answer
71 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 ...
5
votes
2answers
295 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 ...
5
votes
2answers
183 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 ...
5
votes
1answer
70 views

Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...