On Nilary Group Rings

TL;DR

This paper investigates conditions under which group rings are (principally) nilary, establishing relationships between the nilary property of the ring, the group, and the group ring, especially for torsion, abelian, and finite groups.

Contribution

It characterizes when group rings are (principally) nilary, linking properties of the base ring and the group, and provides specific results for finite, abelian, and p-groups.

Findings

If A[G] is nilary, then A is nilary.

For torsion groups, G is a p-group and p is nilpotent in A.

Group algebra over a field of prime characteristic is nilary for finite p-groups.

Abstract

In a ring $A$ an ideal $I$ is called (principally) nilary if for any two (principal) ideals $V, W$ in $A$ with $VW\subseteq I,$ then either $V^n\subseteq I$ or $W^m\subseteq I,$ for some positive integers $m$ and $n$ depending on $V$ and $W;$ a ring $A$ is called (principally) nilary if the zero ideal is a (principally) nilary ideal~\cite{Birkenmeier2013133}. Let $G$ be a group and $A$ be a ring with unity. It is natural to ask when the group ring $A[G]$ is a (principally) nilary ring. We proved that, if $A[G]$ is a (principally) nilary ring, then the ring $A$ is a (principally) nilary ring; also, we proved that if $A[G]$ is a (principally) nilary ring and $G$ is a torsion group, then $A$ is a (principally) nilary ring and $G$ is a $p$-group and $p$ is nilpotent in $A;$ the converse, let $G$ be an abelian or locally finite group, if $A$ is a principally nilary ring and $G$ is a $p$-group and $p$ is nilpotent in $A$ then $A[G]$ is a principally nilary ring. Also, for a finite group $G,$ we proved that, $A[G]$ is a (principally) nilary ring iff $A$ is a (principally) nilary ring and $G$ is a $p$-group and $p$ is nilpotent in $A.$ Finally, we show that if $F$ is a field of prime characteristic $p$ and $G$ is a finite (abelian or locally finite) $p$-group, then the group algebra $F[G]$ is a (principally) nilary ring.