Multiplication modules and homogeneous idealization

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. Homogeneous ideals of R (M) have the form I₍₊N where I is an ideal of R, N a submodule of M and IM ⊆ N. The purpose of this paper is to investigate how properties of a homogeneous ideal I₍₊₎N of R (M) are related to those of I and N. We show that if M is a multiplication R-module and I_(-))N is a meet principal (join principal) homogeneous ideal of R (M) then these properties can be transferred to I and N. We give some conditions under which the converse is true. We also show that I_(-)N is large (small) if and only if N is large in M (I is a small ideal of R).