on the distributivity of the lattice of radical submodules

Let r be a commutative ring with identity and r(rm) de-note 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 n := 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).