Integral Domains in Which Each Ideal Is aW-Ideal

Abstract

ABSTRACT In this paper, we investigate integral domains in which each ideal is a w-ideal (i.e. the d- and w-operations are the same), called the DW-domains. In some sense this study is similar to that one given in Houston and Zafrullah (1988 Houston , E. , Zafrullah , M. (1988). Integral domains in which each t-ideal is divisorial. Michigan Math. J. 35:291–300. [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) [Houston, E., Zafrullah, M. (1988). Integral domains in which each t-ideal is divisorial. Michigan Math. J. 35:291–300.] for the TV-domains. We prove that a domain R is a DW-domain if and only if each maximal ideal of R is a w-ideal, and if R is a domain such that R M is a DW-ideal for each maximal ideal M of R, then so is R, and the equivalence holds when R is v-coherent. We describe the w-operation on pull–backs in order to provide original examples.