Lattice decomposition of modules over commutative rings

Abstract

The direct sum decomposition of a module M in a direct sum $M=M_1\oplus{M_2}$ does not produce, in general, a decomposition of the lattice $L(M)$ of all submodules of $M$ in a direct product of lattices $L(M)=L(M_1)\times{L(M_2)}$. When this happens we say $M=M_1\oplus{M_2}$ is a lattice decomposition of $M$. These particular decompositions have special properties. The main aim of this paper is to characterize, in terms of $\textrm{Supp}(M)$, when the module $M$, over a commutative ring A, have a lattice decomposition. Thus, we show that there is a one-to-one correspondence between lattice decompositions of $M$ and partitions of $\textrm{Supp}(M)$ in two closed under specialization subsets satisfying some extra properties. These extra properties are always satisfied whenever $A$ is noetherian ring; in that case each closed under specialization partition always produces lattice decomposition. In particular, we obtain that a module M such that $A/\textrm{Ann}(M)$ is noetherian has a nontrivial lattice decomposition if, and only if, there exists a partition of the set of all prime ideals, minimal in $\textrm{Supp}(M)$, in two sets $D_1$ and $D_2$. We prove that the lattice decomposition is a local property and also show several applications of the lattice decomposition to the module structure, as well as its behavior in relation to some module constructions, change of ring and ring extensions. On the other hand, if $M$ has a lattice decomposition, then the simple modules which are subfactors of M produce a decomposition of $\sigma[M]$, the category of all modules subgenerated by $M$, in a product of two subcategories.