S-n-ideals of commutative rings

Abstract

Let R be a commutative ring with identity and S a multiplicatively closed subset of R. This paper aims to introduce the concept of S−n-ideals as a generalization of n-ideals. An ideal I of R disjoint with S is called an S−n- ideal if there exists sinS such that whenever abinI for a, binR, then sainsqrt0 or sbinI. The relationships among S−n-ideals, n-ideals, S-prime and S-primary ideals are clarified. Besides several properties, characterizations and examples of this concept, S-n-ideals under various contexts of constructions including direct products, localizations and homomorphic images are given. For some particular S and minN, all S−n-ideals of the ring Zm are completely determined. Furthermore, S−n-ideals of the idealization ring and amalgamated algebra are investigated.