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Fuzzy semirings. (English) Zbl 0796.16038

Summary: We initiate the study of fuzzy semirings and fuzzy \(A\)-semimodules where \(A\) is a semiring and \(A\)-semimodules are representations of \(A\). In particular, semirings all of whose ideals are idempotent, called fully idempotent semirings, are investigated in a fuzzy context. It is proved, among other results, that a semiring \(A\) is fully idempotent if and only if the lattice of fuzzy ideals of \(A\) is distributive, under the sum and product of fuzzy ideals. It is also shown that the set of proper fuzzy prime ideals of a fully idempotent semiring \(A\) admits the structure of a topological space, called the fuzzy prime spectrum of \(A\).

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras
16D10 General module theory in associative algebras
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References:

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