The n-insertive subgroups of units

Abstract

Let R be a finite ring. Let us denote its group of units by G = G[R) and its Jacobson radical by J = J(R). Let n be an arbitrary integer. We prove that R is an n-insertive ring if and only if G is an n-insertive group and show that every n-insertive finite ring is a direct sum of local rings. We prove that if n is a unit, then the local ring R is n-insertive if and only if its Jacobson group 1 + J is n-insertive and find an example to show that this is not true if n is a non-unit.