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Nil-extensions of completely simple semirings. (English) Zbl 1301.16054
From the text: A semiring \(S\) is said to be a quasi completely regular semiring if for any \(a\in S\) there exists a positive integer \(n\) such that \(na\) is completely regular. The present paper is devoted to the study of completely Archimedean semirings. We show that a semiring \(S\) is a completely Archimedean semiring if and only if it is a nil-extension of a completely simple semiring. This result extends the crucial structure theorem of completely Archimedean semigroup.
We show that a semiring is completely Archimedean if and only if it is nil-extension of a completely simple semiring if and only if it is Archimedean and quasi completely regular. The preliminaries and prerequisites we need for this article are discussed in Section 2. In Section 3 we prove some characterization theorems and study few properties of completely Archimedean semirings and finally discuss our main result.

MSC:
16Y60 Semirings
20M10 General structure theory for semigroups
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16S70 Extensions of associative rings by ideals
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