Three Topological Properties from Noetherian Rings

Abstract

The purpose of this paper is to study three concepts that deal with the topologies on ideals of commutative integral domains. We call a domain R prime-injective if for each torsion free R-module M, and all non-zero prime idealscommutes implies that M is injective. From [6, Theorem 1 and the technique of Example 6] this is equivalent to all non-zero ideals of R being open in the topology defined by finite products of non-zero prime ideals as a base of neighborhoods around zero.A domain is strongly prime-injective if for each (torsion theory) topology and for ϕ the set of primes in , ϕ-injective implies -injective for torsion free modules (see [6, 8] for notation). As in the prime-injective case, this is equivalent to being the topology generated by ϕ for all topologies .