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One-sided \(k\)-ideals and \(h\)-ideals in semirings. (English) Zbl 0847.16033

Let \((S,+,\cdot)\) be a semiring with commutative addition. A left ideal \(A\) of \(S\) is said to be a left \(k\)-ideal if \(a+a_1=a_2\) for some \(a_i\in A\) and \(a\in S\) implies \(a\in A\). A left ideal \(A\) of \(S\) is called a left \(h\)-ideal if \(a+a_1+u=a_2+u\) for some \(a_i\in A\) and \(u,a\in S\) implies \(a\in A\). Analogously we define two-sided \(k\)-ideals and \(h\)-ideals of \(S\).
In this paper the authors give sufficient conditions on a semiring \(S\) such that each proper (left) \(k\)-ideal or each proper (left) \(h\)-ideal is contained in a maximal one. They deal with semiring-theoretical generalizations of the fact that ideal \(A\) of a ring \(A\) is a maximal ideal of \(R\) iff the congruence class ring \(R/A\) has only the trivial ideals.

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras
20M12 Ideal theory for semigroups
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