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**On quasi-ideals of semirings.**
*(English)*
Zbl 0799.16036

This paper deals with arbitrary semirings \((S,+,\cdot)\) which may have an absorbing element 0 (defined by \(0a = a0 = 0\) for all \(a \in S\)) or not. A subsemigroup \(Q\) of \((S,+)\) is called a quasi-ideal of \((S,+,\cdot)\) if \(\langle SQ\rangle \cap \langle QS\rangle \subseteq Q\) holds, where \(AB = \{a \cdot b \mid a\in A,\;b\in B\}\) denotes the product of subsets in the semigroup \((S,\cdot)\) and \(\langle C\rangle\) the subsemigroup of \((S,+)\) generated by \(C \subseteq S\). (It is indispensable to distinguish the “products” \(AB\) and \(\langle AB\rangle = \{\sum^ n_{i = 1} a_ ib_ i \mid a_ i \in A,\;b_ i \in B,\;n\in \mathbb{N}\}\) in this context, and the author’s notion just described is much better than that used by the reviewer [in Contributions to general algebra 2, Proc. Klagenfurt Conf. 1982, 375-394 (1983; Zbl 0522.16034)].

The paper is selfcontained and presents various new results on quasi- ideals of semirings. So, e.g., the following statements are equivalent for each semiring \((S,+,\cdot)\): 1) \((S,\cdot)\) is regular. 2) \(RL = L \cap R\) holds for all left ideals \(L\) and right ideals \(R\) of \((S,+,\cdot)\). 3) \(\langle L^ 2\rangle = L\), \(\langle R^ 2\rangle = R\) and \(\langle RL\rangle\) is a quasi-ideal. 4. All quasi-ideals form a regular semigroup with respect to \(\langle Q_ 1 Q_ 2\rangle\). 5) \(Q = \langle Q S Q\rangle\) holds for each quasi-ideal \(Q\). A similar theorem contains equivalent conditions for an element \(s\) of a semiring \((S,+,\cdot)\) to be regular. Other results concern the equations \(Q = \langle Q^ 2\rangle\) and \(\langle Q^ 2\rangle = \langle Q^ 3\rangle\), the interrelation to bi-ideals, the question which quasi- ideals are the intersection of a left and a right ideal, and the investigation of minimal, 0-minimal and canonical quasi-ideals. For instance, if \(L\) and \(R\) are (0-)minimal left and right ideals of \(S\), then either \(\langle RL\rangle = \{0\}\) holds or \(\langle RL\rangle\) is the canonical quasi-ideal \(L \cap R\).

There is one point more of importance. All results are also true if (instead of semirings) one deals with semigroups or rings (the latter clearly considered as rings \((S,+,\cdot)\) and not as semirings, which means that \(\langle C\rangle\) denotes the subgroup of \((S,+)\) generated by \(C\)). These considerations for semigroups and rings are scattered through the literature and often proved by completely different methods. The author has managed to deal with all three cases simultaneously and to give unified proofs, which are no more difficult and even sometimes considerably simpler than the known proofs suitable only for rings or for semigroups.

The paper is selfcontained and presents various new results on quasi- ideals of semirings. So, e.g., the following statements are equivalent for each semiring \((S,+,\cdot)\): 1) \((S,\cdot)\) is regular. 2) \(RL = L \cap R\) holds for all left ideals \(L\) and right ideals \(R\) of \((S,+,\cdot)\). 3) \(\langle L^ 2\rangle = L\), \(\langle R^ 2\rangle = R\) and \(\langle RL\rangle\) is a quasi-ideal. 4. All quasi-ideals form a regular semigroup with respect to \(\langle Q_ 1 Q_ 2\rangle\). 5) \(Q = \langle Q S Q\rangle\) holds for each quasi-ideal \(Q\). A similar theorem contains equivalent conditions for an element \(s\) of a semiring \((S,+,\cdot)\) to be regular. Other results concern the equations \(Q = \langle Q^ 2\rangle\) and \(\langle Q^ 2\rangle = \langle Q^ 3\rangle\), the interrelation to bi-ideals, the question which quasi- ideals are the intersection of a left and a right ideal, and the investigation of minimal, 0-minimal and canonical quasi-ideals. For instance, if \(L\) and \(R\) are (0-)minimal left and right ideals of \(S\), then either \(\langle RL\rangle = \{0\}\) holds or \(\langle RL\rangle\) is the canonical quasi-ideal \(L \cap R\).

There is one point more of importance. All results are also true if (instead of semirings) one deals with semigroups or rings (the latter clearly considered as rings \((S,+,\cdot)\) and not as semirings, which means that \(\langle C\rangle\) denotes the subgroup of \((S,+)\) generated by \(C\)). These considerations for semigroups and rings are scattered through the literature and often proved by completely different methods. The author has managed to deal with all three cases simultaneously and to give unified proofs, which are no more difficult and even sometimes considerably simpler than the known proofs suitable only for rings or for semigroups.

Reviewer: H.J.Weinert (Clausthal-Zellerfeld)

### MSC:

16Y60 | Semirings |

20M12 | Ideal theory for semigroups |

16D25 | Ideals in associative algebras |

12K10 | Semifields |

16D80 | Other classes of modules and ideals in associative algebras |

20M17 | Regular semigroups |