### Semilocal and semiregular group rings

J.B. Srivastava, Sudesh K. Shah
1980 Indagationes Mathematicae (Proceedings)
Throughout this paper a ring will mean an associative ring with identity 1 # 0. The Jacobson radical of a ring R is denoted by J(R). A ring R is semilocal if R/J(R) is Artinian. R is called semiperfect if R/J(R) is Artinian and idempotents can be lifted modulo J(R). An element (I of a ring R is said to be regular (in the sense of von Neumann) if a ra = a for some r E R. If each element of R is regular, R is said to be a regular ring. A ring R is called semiregular if R/J(R) is regular and
more » ... tents can be lifted modulo J(R). Regular rings and semiperfect rings are clearly semiregular. Further it is known that endomorphism rings of injective modules are semiregular. For a more detailed study of semiregular rings and related topics, we refer to  and . Following  we call a ring R left weakly perfect if and only if R satisfies the minimum condition on principal right ideals which are not direct summands. Weakly perfect rings are semiregular [2, Theorem 31. In Section 1, we prove that if RR is a direct summand of RS (where R is a subring of s) and if S is a semilocal ring then R is also a semilocal ring. Some important applications to group rings are given. Semiregular rings and group rings are studied in Section 2 and in Section 3 we have studied weakly perfect group rings. Notation and basic facts involving group rings are taken from  and . 1. In general, subrings of semilocal rings need not be semilocal. Our Theorem 1 below is a useful observation in this direction. It is not hard to see that a ring R is semilocal if and only if J(R) is a finite intersection of maximal right ideals of R. 347