A note on 2-prime and n-weakly 2-prime ideals of semirings

Abstract

We introduce and study the concepts of 2-prime and n-weakly 2-prime (resp. weakly 2-prime) ideals in a commutative semiring. We prove that an integral semidomain S is a valuation semiring if and only if every proper ideal of S is 2-prime and in a principal ideal semidomain the concepts of primary, quasi-primary and 2-prime ideals coincide. We characterize semirings where 2-prime ideals are prime and also characterize semirings where every proper ideal is n-weakly 2-prime (resp. weakly 2-prime)