On the distributivity of the lattice of radical submodules

Let R be a commutative ring with identity and \mathcal{R}(_{R}M) denotes the bounded lattice of radical submodules of an R-module M. We say that M is a radical distributive module, if \mathcal{R}(_{R}M) is a distributive lattice. It is shown that the class of radical distributive modules contains the classes of multiplication modules and finitely generated distributive modules properly. It is shown that if M is a semisimple R-module and for any radical submodule N of M with direct sum complement \tilde{N}, the complementary operation on \mathcal{R}(_{R}M) is defined by N':=\tilde{N}+rad(۰), then \mathcal{R}(_{R}M) with this unary operation forms a Boolean algebra. In particular, if M is a multiplication module over a semisimple ring R, then \mathcal{R}(_{R}M) is a Boolean algebra, which is also a homomorphic image of \mathcal{R}(_{R}R).

کلیدواژه ها:

Radical distributive module ، distributive module ، multiplication module ، semisimple ring ، Boolean algebra homomorphism