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 Γ−semiring of non-negative integers and their properties by defining Euclidean Γ−semiring. A Euclidean Γ−semiring is one of the many special classes of Γ−semirings. Finally, the special class of Γ−semirings discussed in this paper is the class of almost principal ideal Γ−semirings.
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