Lattice decomposition of modules
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International Electronic Journal of Algebra
ModuleRingLatticeLattice decompositionGrothendieck category
García, J. M., Jara, P., & Merino, L. M. (2021). Lattice decomposition of modules. International Electronic Journal of Algebra Volume 30 (2021) 285-303. DOI: [10.24330/ieja.969940]
The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module M produce these decompositions: the lattice decompositions. In a first etage this can be done using endomorphisms of M, which produce a decomposition of the ring EndR(M) as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step we characterize when a particular module M has a lattice decomposition; this can be done, in the commutative case in a simple way using the support, Supp(M), of M; but, in general, it is not so easy. Once we know when a module decomposes, we look for characterizing its decompositions. We show that a good framework for this study, and its generalizations, could be provided by the category sigma[M], the smallest Grothendieck subcategory of Mod - R containing M.