On nil rings satisfying minimum condition on principal right ideals

Abstract

In what follows under a ring we always mean an associative one. The main object of this paper is to describe the structure of nil rings with minimum condition on principal right ideals (Theorem 1). Let Z be the ring of integers. The minimum condition for principal right ideals means that every descending chain of principal right ideals (al),D(a2),~(a8),D ... D terminates after a finite number of steps where (a~),=aiZ+atA. By MHR-ring (MHL-ring) we mean a ring with minimum condition on principal right (left) ideals. MHR-rings were introduced by Sz~sz [7--9] and Faith [3]. Later Bass [2] considered perfect rings, i.e. MHR-rings with unit. Let us recall that a ring A is left T-nilpotent if for any sequence of elements a~ of A, i=1, 2, ..., n, ... there exists a positive integer n such that a~a~as...a,=O. For the basic notions and results of the radical theory we refer to Andrunakievi~--Rjabuhin [1], Szfisz [10], Wiegandt [11]. We denote by