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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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