Ratliff–Rush closures of ideals with respect to a Noetherian module

Let R be a commutative Noetherian ring, E a non-zero finitely generated R-module and I a E-proper ideal of R. The purpose of this paper is to provide some new characterizations of when all powers of I are Ratliff–Rush closed with respect to E and to answer a question raised by W. Heinzer et al. in (The Ratliff–Rush Ideals in a Noetherian Ring: A Survey, in Methods in Module Theory, Dekker, New York, 1992, pp. 149–159).