Abstract
The direct sum decomposition of a module in a direct sum does not produce, in general, a decomposition of the lattice of all submodules of in a direct product of lattices . When this happens we say is a lattice decomposition of . These particular decompositions have special properties: for instance, , and , are complemented distributive submodules.
The main aim of this paper is to characterize, in terms of , when the module , over a commutative ring , have a lattice decomposition. Thus, we show that there is a one-to-one correspondence between lattice decompositions of and partitions of in two closed under specialization subsets satisfying some extra properties. These extra properties are always satisfied whenever is noetherian ring; in that case each closed under specialization partition always produces lattice decomposition. In particular, we obtain that a module such that is noetherian has a nontrivial lattice decomposition if, and only if, there exists a partition of the set of all prime ideals, minimal in , in two sets and such that ideals in are comaximal with ideals in .
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 has a lattice decomposition, then the simple modules which are subfactors of produce a decomposition of , the category of all modules subgenerated by , in a product of two subcategories.
Citation
Josefa M. García. Pascual Jara. Evangelina Santos. "LATTICE DECOMPOSITION OF MODULES OVER COMMUTATIVE RINGS." J. Commut. Algebra 15 (4) 497 - 511, Winter 2023. https://doi.org/10.1216/jca.2023.15.497
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