Principally Injective Rings

✍ Scribed by W.K. Nicholson; M.F. Yousif

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 668 KB
• Volume: 174
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1117

✦ Synopsis

A ring (R) is called right principally injective if every (R)-homomorphism from a principal right ideal to (R) is left multiplication by an element of (R). In this paper various properties of these rings are developed, many extending known results. If, in addition, (R) is semiperfect and has an essential right socle, it is shown: (1) that the right socle equals the left socle, that this is essential on both sides and is finitely generated on the left; (2) that the two singular ideals coincide; and (3) that (R) admits a Nakayama permutation of its basic idempotents. These rings are a natural generalization of the pseudo-Frobenius rings, and our work extends results of Björk and Rutter. We also answer a question of Camillo about commutative principally injective rings in which every ideal contains a uniform ideal. Finally, we show that if the group ring (R G) is principally injective then (R) is principally injective and (G) is locally finite; and that if (R) is right selfinjective and (G) is locally finite then (R G) is principally injective, extending results of Farkas. O 1995 Academic Press, Inc.