On the structure of certain PP-rings

Abstract

All rings considered are associative and have identity elements. By "Artinian rings", "Noetherian rings", "Goldie rings" and "hereditary rings" we shall mean rings which have the respective conditions on both sides. A ring R is a right PP-ring provided every principal right ideal is projective. Small [11], Theorem l, has proved that a right PP-ring which does not certain an infinite collection of orthogonal idempotents is also a left PP-ring. By a PPring we shall mean a ring which is both a right PP-ring and a left PP-ring. We shall call a PP-ring limited if it does not contain an infinite collection of orthogonal idempotents. Chatters [1], Theorem 3.1, has proved that any Noetherian PP-ring can be written as a finite direct sum of prime PP-rings and Artinian PP-rings. In an earlier theorem Levy [10], Theorem 4.3, had proved that any semiprime right Goldie PP-ring is a direct sum of prime rings. The primary purpose of this note is to investigate this theorem of Chatters and to extend it. If I is an ideal of a ring R then by 'CgR(I ) (respectively cg~(I)) we shall mean the set of elements c of R such that whenever reR and rc~I (creI) it follows that r~I. If S is a nonempty subset of R then l(S) and r(S) will denote the left and right annihilators of S, respectively. Our main theorem states that if R is a right Noetherian PP-ring with prime radical N then R is the direct sum of a semiprime ring and a ring with essential right socle if and only if '~(O)c_'Cg(K), where K =Nnl(N) (Theorem 3.7). A rather similar argument shows that if R is a PP-ring with prime radical N such that R/N is a Goldie ring and N is finitely generated both as a right ideal and as a left ideal then R is the direct sum of a semiprime ring and an Artinian ring (Theorem 4.3). Fuelberth and Kusmanovich [3], Theorem 3.12, have proved that a hereditary ring which has a semiprimary right and left quotient ring can be written as a finite direct sum of semiprimary rings and prime rings. Recall the Chatters in [1], Theorem4.1, has shown that a PP-ring R with prime r ad i ca lN has a semiprimary right quotient ring if and only if R/N is a right Goldie ring and