Lattice decomposition of modules García Hernández, Josefa María Jara Martínez, Pascual Merino González, Luis Miguel Module Ring Lattice Lattice decomposition Grothendieck category 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. 2021-09-16T12:12:48Z 2021-09-16T12:12:48Z 2021 info:eu-repo/semantics/article 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] http://hdl.handle.net/10481/70239 10.24330/ieja.969940 eng http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess Atribución 3.0 España International Electronic Journal of Algebra