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Results tagged with ra.rings-and-algebras
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user 19075
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.
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 …
- 3,738
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 $\ …
- 3,738
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 $n$ …
- 3,738
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 …
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 …
- 3,738
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 …
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