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Regular additively inverse semirings. (English) Zbl 1160.16309

Summary: We show that in a regular additively inverse semiring \((S,+,\cdot)\) with 1 satisfying the conditions (A) \(a(a+a')=a+a'\), (B) \(a(b+b')=(b+b')a\) and (C) \(a+a(b+b')=a\), for all \(a,b\in S\), the sum of two principal left ideals is again a principal left ideal. Also, we decompose \(S\) as a direct sum of two mutually inverse ideals.

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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