LATTICE DECOMPOSITION OF MODULES

Authors : J M GARCIA, P JARA, L M MERINO

Pages : 285-303

Doi:10.24330/ieja.969940

View : 11 | Download : 15

Publication Date : 2021-07-17

Article Type : Research Paper

Abstract :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 \emph{lattice decompositions}. In a first \textit{\`{e}tage} this can be done using endomorphisms of $M$, which produce a decomposition of the ring $\End_Rinsert ignore into journalissuearticles values(M);$ 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, $\Suppinsert ignore into journalissuearticles values(M);$, 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 $\rMod{R}$ containing $M$.
Keywords : Module, ring, lattice, lattice decomposition, Grothendieck category