On Distinguishing Local Finite Rings from Finite Rings Only by Counting Elements and Zero Divisors

Marcos José González

Abstract

Thepurposeof this shortcommunication is to prove thefollowing: {\emLet$R$ be a finite associativeringwith unit. Then$R$ is local if and only if $|R| = p^n$ and $|Z(R)| = p^m$for some prime number$p$ and integers$1\leq m <n$}.  Forthecommutative case, this havebeenrecentlydiscovered byBehboodi and Beyranvand\cite[Theorem~3]{bb}. Wewill also presentyetanother proof for thecommutative case.

Keywords

Finite ring, Zero-divisor, Local rings

Full Text:

PDF