On Quasi-Harada Rings

✍ Scribed by Yoshitomo Baba; Kenichi Iwase

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 299 KB
• Volume: 185
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0339

✦ Synopsis

the following two conditions: Ž . * Every non-small left R-module contains a non-zero injective submodule. Ž . * * Every non-cosmall right R-module contains a non-zero projective direct summand. Ž . K. Oshiro Hokkaido Math. J. 13, 1984, 310᎐338 further studied the above Ž . Ž 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 Ž . Ž . ideals with * * a right co-Harada ring abbreviated right co-H-ring . K. Oshiro Ž . Math. J. Okayama Uni¨. 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 ring R is a left H-ring if and only if R is a perfect ring and for any left non-small primitive idempotent e of R there exists a non-negative integer t such that e Ž . Ž . Ä 4 a RerS Re is injective for any k g 0, . . . , t and R kR e Ž . Ž . Ž . b RerS R e is a small module, where S Re denotes the k-th socle R tq 1R k R e of the left R-module Re.