On the distributivity of the lattice of radical submodules

Let R be a commutative ring with identity and R(RM) denote the bounded lattice of radical submodules of an R-module M. We say that M is a radical distributive module, if R(RM) 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. Also, it is shown that if M is a radical distributive semisimple R-module and for any radical submodule N of M with direct sum complement N˜, the complementary operation on R(RM) is defined by N0 := N˜ + rad{0}, then R(RM) with this unary operation forms a Boolean algebra. In particular, if M is a multiplication module over a semisimple ring R, then R(RM) is a Boolean algebra, which is also a homomorphic image of R(RR)