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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 $n\in\bbfN$ 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.

16E50Von Neumann regular rings and generalizations
16S50Endomorphism rings: matrix rings