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Γ-semirings. I. (English) Zbl 0846.16034

Let M and Γ be additive abelian semigroups with identity elements 0 and 0 ' respectively. If there exists a mapping M×Γ×MM (images to be denoted xγy, x,yM, γΓ) satisfying for all x,y,zM, γ,μΓ: (a) xγ(yμz)=(xγy)μz (b) xγ(y+z)=xγy+xγz; (x+y)γz=xγz+yγz; x(γ+μ)y=xγy+xμy (c) xγ0=0γx=x0 ' y=0 then M is called a Γ-semiring. Numerous examples are given in the paper, and various concepts, analogous to those for semirings are defined. A Γ-semiring M is called regular (resp. strongly regular) if for all aM there exist b i M, α i ,β i Γ such that a= i=1 n aα i b i β i a (resp. there exist bM, α,βΓ such that a=aαbβa). The center of M is the set {aMaαx=xαaxM,αΓ}. An element a of M is nilpotent if for each xM, γΓ, there exists n such that (aγ) n a=0. a is idempotent if there exists αΓ such a=aαa. If M is a Γ-semiring, the set M mn of m×n matrices with entries from M is a Γ nm -semiring with the natural operations of matrix addition and multiplication.

The following results are proved: The center B of a strongly regular Γ-semiring is a strongly regular sub Γ-semiring of M. Let M be a strongly regular Γ-semiring. If all the idempotent elements of M are in its center, then M has no nonzero nilpotent elements. – The matrix Γ nm -semiring M mn is regular if and only if M is a regular Γ-semiring.


MSC:
16Y60Semirings
16E50Von Neumann regular rings and generalizations
16S50Endomorphism rings: matrix rings