On quasi-ideals of semirings

Christoph Dönges

doi:10.1155/s0161171294000086

Abstract

Several statements on quasi-ideals of semirings are given in this paper, where these semirings may have an absorbing elementOor not. In Section 2 we characterize regular semirings and regular elements of semi-rings using quasi-ideals (cf. Thms. 2.1, 2.2 and 2.7). In Section 3 we deal with(O−)minimal and canonical quasi-ideals. In particular, if the considered semiringSis semiprime or quasi-reflexive, we present criterions which allow to decide easily whether an(O−)minimal quasi-ideal ofSis canonical (cf. Thms. 3.4 and 3.8). IfSis an arbitrary semiring, we prove that for(O−)minimal left and right idealsLandRofSthe product〈RL〉⫅L⋂Ris either{O}or a canonical quasi-ideal ofS(Thm. 3.9). Moreover, for each canonical quasi-idealQof a semiringSand each elementa∈S,Qais either{O}or again a canonical quasi-ideal ofS(Thm. 3.11), and the product〈Q1Q2〉of canonical quasi-idealsQ1,Q2ofSis either{O}or again a canonical quasi-ideal ofS(Thm. 3.12). Corresponding results to those given here for semirings are mostly known as well for rings as for semigroups, but often proved by different methods. All proofs of our paper, however, apply simultaneously to semirings, rings and semigroups (cf. Convention 1.1), and we also formulate our results in a unified way for these three cases. The only exceptions are statements on semirings and semigroups without an absorbing elementO, which cannot have corresponding statements on rings since each ring has its zero as an absorbing element.

Highlights

Several statements on quasi-ideals of semirings are given in this paper, where these semirings may have an absorbing element O or not
Convention 1.1), and we formulate our results in a unified way for these three cases
PROOF. a) By Theorem 2.1 each quasi-ideal Q of S has the intersection property Q (SQ> N (QS) (Q), N (Q)r L N R and condition (2) of Theorem 2.1 implies Q L R (RL). b) For each quasi-ideal Q of S it follows by Theorem 2.1 (4), that (Q) is a quasi-ideal of S and that there is a quasi-ideal X of S such that which yields (Q2) (Q3) since (Q) c_ (Q) is clear. c) Since (SB> is a left ideal and (BS) is a right idea/of S, Theorem 2.1 (2) impnes (SB) c (BS) (BSSB) c_ (BSB) c_B, so B is a quasi-ideal of S. d) By Lemma 2.3 the two-sided ideal T of S is a regular sub(semi)ring or subsemigroup of S

Summary

Introduction

Several statements on quasi-ideals of semirings are given in this paper, where these semirings may have an absorbing element O or not. Let S be a semiring, a ring or a semigroup, L a left and R a right ideal of S 2. QUASI-IDEALS AND REGULARITY Let S be a semiring, a ring or a semigroup. S of a semiring, a ring or a semigroup S is regular iff one of the following statements holds"

Results

Conclusion