On complete lattices of radical submodules and z-submodules

Abstract

Let M be a module over a commutative ring R, andR(RM) denote the complete lattice of radical submodules of M. It is shown that if M is a multiplication R-module, then R(RM) is a frame. In particular, if M is a finitely generated multiplication R-module, then R(RM) isa coherent frame and if, in addition, M is faithful, then the assignment N→(N : M)z defines a coherent map from R(RM) to the coherent frame Z(RR)of z-ideals of R. As a generalization of z-ideals, a proper submodule N of M is called a z-submodule of M if for any x ∈ M and y ∈ N such that every maximal submodule of M containing y also contains x,thenx ∈ N. The set of z-submodules of M, denoted Z(RM), forms a complete lattice with respect to the order of inclusion. It is shown that if M is a finitely generated faithful multiplication R-module, then Z(RM) is a coherent frame and the assignment N→ Nz (where Nz is the intersectionofall z-submodules of M containing N) is a surjective coherent map from R(RM)toZ(RM). In particular, in this case, R(RM) is a normal frame if and only if Z(RM) is a normal frame.