On Quasi-Harada Rings

Abstract

Listen

Translate article

M. Harada (“Ring Theory, Proceedings of 1978 Antwerp Conference,” pp. 669–690, Dekker, New York, 1979) studied the following two conditions:(*) Every non-small leftR-module contains a non-zero injective submodule.(*)* Every non-cosmall rightR-module contains a non-zero projective direct summand.K. Oshiro (Hokkaido Math. J.13, 1984, 310–338) further studied the above conditions, and called a left artinian ring with (*) a left Harada ring (abbreviated left H-ring) and a ring satisfying the ascending chain condition for right annihilator ideals with (*)* a right co-Harada ring (abbreviated right co-H-ring). K. Oshiro (Math. J. Okayama Univ.31, 1989, 161–178) showed that left H-rings and right co-H-rings are the same rings. Here we are particularly interested in the following characterization of a left H-ring given in Harada's paper above: A ringRis a left H-ring if and only ifRis a perfect ring and for any left non-small primitive idempotenteofRthere exists a non-negative integertesuch that(a)RRe/Sk(RRe) is injective for anyk∈{0,…,te} and(b)RRe/Ste+1(RRe) is a small module, whereSk(RRe) denotes thek-th socle of the leftR-moduleRe.This characterization implies(+)Ste+1(RRe) is a uniserial leftR-module for any left non-small primitive idempotentein a left H-ringR.In this paper, we generalize left H-rings by removing (+). Concretely, since a left H-ringRis also characterized by the statement thatRis left artinian and for any primitive idempotentgofRthere exist a primitive idempotentegofRand a non-negative integerkgsuch that the injective hull of the leftR-moduleRg/Jgis isomorphic toRe/Skg(RReg), whereJis the Jacobson radical ofRR, we define a more general class of rings by the condition that for any primitive idempotentgofRthere exists a primitive idempotenteofRsuch that the injective hull of the leftR-moduleRg/Jgis isomorphic toRe/{x∈Re|gRx=0}, and call it a left quasi-Harada ring (abbreviated left QH ring). In Section 1 we characterize a left QH ring by generalizing the characterizations of a left H-ring given by M. Harada and K. Oshiro. We also consider the weaker rings, left QF-2 rings and right QF-2* rings. K. Oshiro (Math. J. Okayama Univ.32, 1990, 111–118) described the connection between left H-rings and QF rings. In Section 2 we describe the connection between two-sided QH rings and QF rings.