Let and be additive abelian semigroups with identity elements 0 and respectively. If there exists a mapping (images to be denoted , , ) satisfying for all , : (a) (b) ; ; (c) then is called a -semiring. Numerous examples are given in the paper, and various concepts, analogous to those for semirings are defined. A -semiring is called regular (resp. strongly regular) if for all there exist , such that (resp. there exist , such that ). The center of is the set . An element of is nilpotent if for each , , there exists such that . is idempotent if there exists such . If is a -semiring, the set of matrices with entries from is a -semiring with the natural operations of matrix addition and multiplication.
The following results are proved: The center of a strongly regular -semiring is a strongly regular sub -semiring of . Let be a strongly regular -semiring. If all the idempotent elements of are in its center, then has no nonzero nilpotent elements. – The matrix -semiring is regular if and only if is a regular -semiring.