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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.

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