Questions tagged [noncommutative-algebra]

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

In abstract algebra, the theory of rings which are not necessarily commutative is called "noncommutative algebra." In this way it is a generalization of commutative algebra. Some results from commutative algebra hold in noncommutative algebra, but many results break down.

The ring of quaternions was among the first motivating examples of noncommutative rings. Other familiar examples include the $n\times n$ matrix ring over any ring ($n>1$).

A few examples of some differences between commutative and noncommutative algebra:

  • If $R$ is a commutative ring, and $R^n\cong R^m$ as $R$ modules for some positive integers $m$ and $n$, then $m=n$. In contrast, there is a noncommutative ring such that $R^m\cong R^n$ for every pair of positive integers $m,n$.

  • Any commutative ring without zero divisors can be embedded in a field. There are examples of noncommutative rings without zero divisors which cannot be embedded into a division ring. This is one of many signs that show localization does not work well for many noncommutative rings.

  • The module $R_R$ may have different properties from the module $_RR$. For one thing, one could be Noetherian (or Artinian) without the other having the same property.

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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. However, the author noted that the construction…
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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 examples of such rings but there are no answers.…
user23211
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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.
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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 $R\neq \emptyset$ be a set, let $\oplus:A\times A \to A$ and…
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(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 with physical chemistry: cond-mat, fluids, QM of…
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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 is taken over all finite, non-commutative rings,…
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Is a finitely generated projective module a direct summand of a *finitely generated* free module?

Let $R$ be a (not necessarily commutative) ring and $P$ a finitely generated projective $R$-module. Then there is an $R$-module $N$ such that $P \oplus N$ is free. Can $N$ always be chosen such that $P \oplus N$ is free and finitely…
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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 idea $$ (a + ib + jc) (a -ib -jc) =…
asmaier
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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.
pardis
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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 appeared on math.stackexchange a couple of times.…
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Do people ever study non-commutative fields?

I've heard of a field, and I've heard of a non-commutative (or "not-necessarily commutative) rings. Do people ever study non-commutative fields?
goblin GONE
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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 over general unital rings. For example, suppose $R$…
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Why is the constant term of $(1+x+y+xy)^n$ equal to $\frac{1}{2}\binom{2n}{n}$?

If we define this: for any $x,y$ such that $x^2=y^2=1,xy\neq yx$, express in terms of $n$ the constant term of the expression $$f_{n}=(1+x+y+xy)^n\,.$$ I guess this result is $\dfrac{1}{2}\binom{2n}{n}$. for $n=1$, we have $f_{1}=1+x+y+xy$ the…
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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 Analysis (page 435). Let…
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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 over a ring $A$). Does the coproduct even exist…
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