# Turkish Journal of Mathematics

## DOI

10.55730/1300-0098.3412

## Abstract

A ring $A$ is (principally) nilary, denoted (pr-)nilary, if whenever $XY=0,$ then there exists a positive integer $n$ such that either $X^n=0$ or $Y^n=0$ for all (principal) ideals $X$, $Y$ of $A$. We determine necessary and/or sufficient conditions for the group ring $A[G]$ to be (principally) nilary in terms of conditions on the ring $A$ or the group $G$. For example, we show that: (1) If $A[G]$ is (pr-)nilary, then $A$ is (pr-)nilary and either $G$ is prime or the order of each finite nontrivial normal subgroup of $G$ is nilpotent in $A$. (2) Assume that $G$ is finite. Then $G$ is nilpotent and $A[G]$ is (pr-)nilary if and only if $G$ is a $p$-group, $char(A)=p^\alpha $ ($p$ is a prime), and $A$ is (pr-)nilary. (3) Let $G$ be a finite supersolvable group such that $q$ is the smallest prime dividing $ G ,$ and $F$ is a field with $char(F)=q$. Then $F[G]$ is nilary if and only if $G$ is a $q$-group. Examples are provided to illustrate and delimit our results.

## Keywords

Nilary ring, indecomposable ring, group ring, group algebra, quasi-Frobenius ring

## First Page

1051

## Last Page

1072

## Recommended Citation

AL-MALLAH, OMAR; BIRKENMEIER, GARY; and ALNOGASHI, HAFEDH
(2023)
"Nilary group rings and algebras,"
*Turkish Journal of Mathematics*: Vol. 47:
No.
4, Article 2.
https://doi.org/10.55730/1300-0098.3412

Available at:
https://journals.tubitak.gov.tr/math/vol47/iss4/2