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 the group $A[G]$ to be nilary in terms on $G$. For example, we show that: (1) If (pr-)nilary and $G$ prime order each finite nontrivial normal subg...