On divisor topology of modules over domains

Abstract

Let M be a module over a domain R and M#={0 not equal m is an element of M:Rm not equal M} be the set of all nonzero nongenerators of M. Consider the following equivalence relation similar to on M# given by m similar to n if and only if Rm=Rn for every m,n is an element of M#. Let EC(M#) be the set of all equivalence classes of M# with respect to similar to. In this paper, we construct a topology on EC(M#) which is called the divisor topology of M and is denoted by D(M). Actually, D(M) is an extension of the divisor topology D(R) over domains to modules in the sense of Yi & gbreve;it and Ko & ccedil;. We investigate separation axioms Ti for every 0 <= i <= 5, first and second countability, connectivity, compactness, nested property, and Noetherian property on D(M). Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of D(M). Furthermore, we prove that D(M) is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when D(M) is a discrete space.