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−)mini...
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Wiley
1994-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171294000086 |
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| author | Christoph Dönges |
| author_facet | Christoph Dönges |
| author_sort | Christoph Dönges |
| collection | DOAJ |
| description | 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 〈RL〉⫅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, Qa is either {O} or again a canonical quasi-ideal of S (Thm. 3.11), and the product 〈Q1Q2〉 of canonical quasi-ideals Q1, Q2 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. |
| format | Article |
| id | doaj-art-17d017c777d346c9b3cf6b8c65b69376 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1994-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-17d017c777d346c9b3cf6b8c65b693762025-02-03T01:26:49ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251994-01-01171475810.1155/S0161171294000086On quasi-ideals of semiringsChristoph Dönges0Institut für Mathematik, Technische Universität Clausthal, Erzstraße 1, Clausthal-Zellerfeld DW-3392, GermanySeveral 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 〈RL〉⫅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, Qa is either {O} or again a canonical quasi-ideal of S (Thm. 3.11), and the product 〈Q1Q2〉 of canonical quasi-ideals Q1, Q2 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.http://dx.doi.org/10.1155/S0161171294000086quasi-idealsregular elementsregular semirings(O−)minimal quasi-idealscanonical quasi-ideals. |
| spellingShingle | Christoph Dönges On quasi-ideals of semirings International Journal of Mathematics and Mathematical Sciences quasi-ideals regular elements regular semirings (O−)minimal quasi-ideals canonical quasi-ideals. |
| title | On quasi-ideals of semirings |
| title_full | On quasi-ideals of semirings |
| title_fullStr | On quasi-ideals of semirings |
| title_full_unstemmed | On quasi-ideals of semirings |
| title_short | On quasi-ideals of semirings |
| title_sort | on quasi ideals of semirings |
| topic | quasi-ideals regular elements regular semirings (O−)minimal quasi-ideals canonical quasi-ideals. |
| url | http://dx.doi.org/10.1155/S0161171294000086 |
| work_keys_str_mv | AT christophdonges onquasiidealsofsemirings |