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\(\Gamma\)-semirings. I. (English) Zbl 0846.16034
Let \(M\) and \(\Gamma\) be additive abelian semigroups with identity elements 0 and \(0'\) respectively. If there exists a mapping \(M\times\Gamma\times M\to M\) (images to be denoted \(x\gamma y\), \(x,y\in M\), \(\gamma\in\Gamma\)) satisfying for all \(x, y, z\in M\), \(\gamma,\mu\in\Gamma\): (a) \(x\gamma (y\mu z)=(x\gamma y)\mu z\) (b) \(x\gamma (y+z)=x\gamma y+x\gamma z\); \((x+y)\gamma z=x\gamma z+y\gamma z\); \(x (\gamma+\mu) y=x\gamma y+x\mu y\) (c) \(x\gamma 0=0\gamma x=x 0' y=0\) then \(M\) is called a \(\Gamma\)-semiring. Numerous examples are given in the paper, and various concepts, analogous to those for semirings are defined. A \(\Gamma\)-semiring \(M\) is called regular (resp. strongly regular) if for all \(a\in M\) there exist \(b_i\in M\), \(\alpha_i,\beta_i\in\Gamma\) such that \(a=\sum^n_{i =1} a\alpha_i b_i\beta_i a\) (resp. there exist \(b\in M\), \(\alpha,\beta\in\Gamma\) such that \(a=a\alpha b\beta a\)). The center of \(M\) is the set \(\{a\in M\mid a\alpha x=x\alpha a\;\forall x\in M,\;\alpha\in\Gamma\}\). An element \(a\) of \(M\) is nilpotent if for each \(x\in M\), \(\gamma\in\Gamma\), there exists \(n\in\mathbb{N}\) such that \((a\gamma)^n a=0\). \(a\) is idempotent if there exists \(\alpha\in\Gamma\) such \(a=a\alpha a\). If \(M\) is a \(\Gamma\)-semiring, the set \(M_{mn}\) of \(m\times n\) matrices with entries from \(M\) is a \(\Gamma_{nm}\)-semiring with the natural operations of matrix addition and multiplication.
The following results are proved: The center \(B\) of a strongly regular \(\Gamma\)-semiring is a strongly regular sub \(\Gamma\)-semiring of \(M\). Let \(M\) be a strongly regular \(\Gamma\)-semiring. If all the idempotent elements of \(M\) are in its center, then \(M\) has no nonzero nilpotent elements. – The matrix \(\Gamma_{nm}\)-semiring \(M_{mn}\) is regular if and only if \(M\) is a regular \(\Gamma\)-semiring.

16Y60 Semirings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16S50 Endomorphism rings; matrix rings