Multiplicative closure operations on ring extensions

Abstract

Let A ⊆ B be a ring extension and G be a set of A -submodules of B . We introduce a class of closure operations on G (which we call multiplicative operations on ( A , B , G ) ) that generalizes the classes of star, semistar and semiprime operations. We study how the set Mult ( A , B , G ) of these closure operations varies when A , B or G vary, and how Mult ( A , B , G ) behaves under ring homomorphisms . As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.