On Nilary Group Rings

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.