Abstract

The generalization of the ring of ordinary integers and their properties into an Euclidean ring is well known. Every ideal in an Euclidean ring is a principal ideal, as is also widely known. That is, the Euclidean ring is a principal ideal ring. This paper aims to generalize the $\Gamma-$semiring of non-negative integers and their properties by defining Euclidean $\Gamma-$semiring. A Euclidean $\Gamma-$semiring is one of the many special classes of $\Gamma-$semirings. Finally, the special class of $\Gamma-$semirings discussed in this paper is the class of almost principal ideal $\Gamma-$semirings.

Keywords and Phrases

Euclidean $\Gamma-$semiring, Almost Principal ideals and Almost Principal ideal $\Gamma-$semirings.

A.M.S. subject classification

16Y60, 16U40, 46J20.

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