On complete lattices of radical submodules and \( z \)-submodules

Let M be a module over a commutative ring R, and \(\mathcal {R}(_{R}M)\) denote the complete lattice of radical submodules of M. It is shown that if M is a multiplication R-module, then \(\mathcal {R}(_{R}M)\) is a frame. In particular, if M is a finitely generated multiplication R-module, then \(\mathcal {R}(_{R}M)\) is a coherent frame and if, in addition, M is faithful, then the assignment \(N\mapsto (N:M)_{ z }\) defines a coherent map from \(\mathcal {R}(_{R}M)\) to the coherent frame \(\mathcal {Z}(_{R}R)\) 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 \(x\in M\) and \(y\in N\) such that every maximal submodule of M containing y also contains x, then \(x\in N\). The set of \( z \)-submodules of M, denoted \(\mathcal {Z}(_{R}M)\), 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 \(\mathcal {Z}(_{R}M)\) is a coherent frame and the assignment \(N\mapsto N_{ z }\) (where \(N_{ z }\) is the intersection of all \( z \)-submodules of M containing N) is a surjective coherent map from \(\mathcal {R}(_{R}M)\) to \(\mathcal {Z}(_{R}M)\). In particular, in this case, \(\mathcal {R}(_{R}M)\) is a normal frame if and only if \(\mathcal {Z}(_{R}M)\) is a normal frame.