Tagged Questions

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

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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 ...
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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. ...
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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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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...