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

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