On quasi-ideals of semirings

Several statements on quasi-ideals of semirings are given in this paper, where these semirings may have an absorbing element O or 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 semiring S is semiprime or quasi-reflexive, we present criterions which allow to decide easily whether an ( O − ) minimal quasi-ideal of S is canonical (cf. Thms. 3.4 and 3.8). If S is an arbitrary semiring, we prove that for ( O − ) minimal left and right ideals L and R of S the product 〈 R L 〉 ⫅ L ⋂ R is either { O } or a canonical quasi-ideal of S (Thm. 3.9). Moreover, for each canonical quasi-ideal Q of a semiring S and each element a ∈ S , Q a is either { O } or again a canonical quasi-ideal of S (Thm. 3.11), and the product 〈 Q 1 Q 2 〉 of canonical quasi-ideals Q 1 , Q 2 of S is either { O } or again a canonical quasi-ideal of S (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 element O , which cannot have corresponding statements on rings since each ring has its zero as an absorbing element.