On complete lattices of radical submodules and $$ z $$-submodules - Moghimi, Hosein Fazaeli; Mohebian, Seyedeh Fatemeh (Algebra universalis, 2025)
Let M be a module over a commutative ring R, and R(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) is a 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, then x∈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 intersection of all z-submodules of M containing N) is a surjective coherent map from R(RM) to Z(RM). In particular, in this case, R(RM) is a normal frame if and only if Z(RM) is a normal frame.