Multiplicative closure operations on ring extensions

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.