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${\Gamma }$-semirings. I. (English) Zbl 0846.16034

Let $M$ and ${\Gamma }$ be additive abelian semigroups with identity elements 0 and ${0}^{\text{'}}$ respectively. If there exists a mapping $M×{\Gamma }×M\to M$ (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) $x\gamma \left(y\mu z\right)=\left(x\gamma y\right)\mu z$ (b) $x\gamma \left(y+z\right)=x\gamma y+x\gamma z$; $\left(x+y\right)\gamma z=x\gamma z+y\gamma z$; $x\left(\gamma +\mu \right)y=x\gamma y+x\mu y$ (c) $x\gamma 0=0\gamma x=x{0}^{\text{'}}y=0$ 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 $a={\sum }_{i=1}^{n}a{\alpha }_{i}{b}_{i}{\beta }_{i}a$ (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 $\left\{a\in M\mid a\alpha x=x\alpha a\phantom{\rule{4pt}{0ex}}\forall x\in M,\phantom{\rule{4pt}{0ex}}\alpha \in {\Gamma }\right\}$. An element $a$ of $M$ is nilpotent if for each $x\in M$, $\gamma \in {\Gamma }$, there exists $n\in ℕ$ such that ${\left(a\gamma \right)}^{n}a=0$. $a$ is idempotent if there exists $\alpha \in {\Gamma }$ such $a=a\alpha a$. If $M$ is a ${\Gamma }$-semiring, the set ${M}_{mn}$ of $m×n$ 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 16S50 Endomorphism rings: matrix rings