Von Neumann Regularity in Semirings

✍ Scribed by H. Subramanian

• Publisher: John Wiley and Sons
• Year: 1970
• Tongue: English
• Weight: 370 KB
• Volume: 45
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19700450104

✦ Synopsis

We study iii this note the notion of (von NEUMANN) regularity in commutative semirings (known also as commutative halfrings). The results of [Ill form the main inspiration and basis. Certain equivalent conditions of regularity in rings (commutative with unit) are essentially due to the fact that an idempotent in a ring is also a -'complemented element" (the definition follows). I n this light, we observe that semirings are a generalization of rings and distributive lattices. This has some pathologies [12]; but up to the point of this paper, the discussions on complemented elements in [I21 are useful.

By a semiring, we mean a set endowed with two commutative and associative binary operations + and such that . i s distributed by + . We assume throughout that a semiring has neutral elements 0 and 1 for + and a respectively and has the property that 0 x = 0. A complemented element P in a semiring is an element such that there exists a "complement" .f satisfying e + f = 1 and e f = 0. The usual terminologies in the theory of