Multiplication modules and homogeneous idealization II

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M. Homogenous ideals of R(M) have the form I₍₊₎N, where I is an ideal of R and N a submodule of M such that IM ⊆ N. A ring R (M) is called a homogeneous ring if every ideal of R (M) is homogeneous. In this paper we continue our recent work on the idealization of multiplication modules and give necessary and sufficient conditions for a homogeneous ideal to be an almost (generalized, weak) multiplication, projective, finitely generated flat, pure or invertible (q-invertible). We determine when a ring R(M) is a general ZPI-ring, distributive ring, quasi-valuation ring, P-ring, coherent ring or finite conductor ring. We also introduce the concept of weakly prime submodules generalizing weakly prime ideals. Various properties and characterizations of weakly prime submodules of faithful multiplication modules are considered.