Ideal theory in graded semirings

Abstract

An A-semiring has commutative multiplication and the property that every proper ideal B is contained in a prime ideal P , with p B, the intersection of all such prime ideals. In this paper, we define homogeneous ideals and their radicals in a graded semiring R. When B is a proper homogeneous ideal in an A-semiring R, we show that p B is homogeneous whenever p B is a k-ideal. We also give necessary and sufficient conditions that a homogeneous k-ideal P be completely prime (i.e., F 62 P;G 62 P implies FG 62 P ) in any graded semiring. Indeed, we may restrict F and G to be homogeneous elements of R.