IDEALS AND ALMOST PRINCIPAL IDEALS IN EUCLIDEAN $\Gamma-$SEMIRINGS

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.