A scheme associated to modules over commutative rings

Abstract

For any moduleMover a commutative ring R with identity,we consider Sch(M) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch(M) and introduce a sheaf OSch(M) of commutative rings on it, which makes (Sch(M),OSch(M)) into a scheme. In particular, if M is a module over a Noetherian ring R, then (Sch(M),OSch(M)) is a locally Noetherian scheme. Among others,we give sufficient conditions for Sch(M) to be an affine scheme.