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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

2 votes

Why do we associate a graph to a ring?

The Fischer graph is one of examples; see page 569 of Suzuki, Michio. Group theory. II. Translated from the Japanese. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemati …
1 vote
0 answers
71 views

Non-zero homomorphism from a module to its ground ring

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\ …
Alireza Abdollahi's user avatar
6 votes
1 answer
355 views

Zero divisors in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ suc …
Alireza Abdollahi's user avatar
6 votes
0 answers
219 views

Is always the ratio (number of commuting pairs of elements in a ring or a Lie algebra)/(the ...

(1) Is there a finite nilpotent ring $R$ such that the ratio $$c(R)=\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$ is not integer? Edit 1: The nilpotent condition is put later. Edit/Answer: A …
Alireza Abdollahi's user avatar
8 votes
Accepted

Centralizer of a Matrix over a Finite Field

Let me add some cases in which one has a clear answer: [R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991., Corollary 4.4.18]. Let $F$ be a field and $n$ …
Alireza Abdollahi's user avatar
51 votes
Accepted

Invertible matrices over noncommutative rings

See: R.N. Gupta, Anjana Khurana, Dinesh Khurana, and T.Y. Lam, Rings over which the transpose of every invertible matrix is invertible; J. Algebra 322 (2009), no. 5, 1627–1636 (MR). Abstract: We pr …
Alireza Abdollahi's user avatar