Lifting Idempotents in Associative Pairs

✍ Scribed by Miguel Angel Fortes Escalona; Inmaculada de las Peñas Cabrera; Esperanza Sánchez Campos

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 115 KB
• Volume: 222
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8025

✦ Synopsis

In 1977, Nicholson developed the theory of suitable rings Trans. Amer. Math. Ž . . Soc. 229 1977 , 269᎐278 , namely, those in which idempotents lift modulo every Ž . left right ideal. In this paper the concepts of lifting idempotents modulo every left Ž q y . Ž . ideal in an associative pair A , A and lifting von Neumann regular elements q Ž y . modulo every left ideal of A resp. A are introduced and shown to be equivalent. We study the behavior of a pair and its standard embedding with respect to the property of being idempotent-lifting. It is also proved that the Jacobson radical can be characterized as the largest one-sided ideal containing no nonzero regular elements. Finally, we show that lifting orthogonal idempotents is possible in this class of pairs.