Open AccessSOME REMARKS ON THE FINITENESS CONDITIONS OF RINGS
Author(s) -
Asmaa Kamal
Publication year - 1991
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.22.1991.4594
Subject(s) - mathematics , ideal (ethics) , idempotence , minimal ideal , abelian group , injective function , pure mathematics , ring (chemistry) , maximal ideal , discrete mathematics , combinatorics , law , chemistry , organic chemistry , political science
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 $A[x]$.