2021/02/01 by García, Josefa M., Jara, Pascual, Merino, Luis M.
#13C60 #16D70 #18E10 #18E35 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2102.01179
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 étage this can be done using endomorphisms of M, which produce a decomposition of the ring \textrmEndR(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, \textrmSupp(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 σ[M], the smallest Grothendieck subcategory of Mod-R containing M.