SOME REMARKS ON THE FINITENESS CONDITIONS OF RINGS

The aim of this paper is to study the finiteness of rings. We prove that if $A$ is a regular left-self-injective ring, then $A$ is of type III (purely infinite) implies that $E(A[x])$ is, and $A$ contains an abelian idempotent if and only if $E(A[x])$ contains an abelian idempotent. Also we prove that. If $A$ is a regular left self-injective ring and $J$ is a left ideal in $A[x]$ such that $C(J)$ is an essential left ideal in $A$, then there exists a countably generated left ideal $J'$ in $A[x]$ such that $C(J')$ is an essential left ideal in $A$, and if $J'$ is an essential left ideal in $A[x]$, then $J$ is an essential left ideal in $A[x]$.