Multiplication Modules and Homogeneous Idealization Iv

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 . Homogeneuous ideals of R(M) have the form I(+)N where I is an ideal of R, N a submodule of M such that IM ⊆ N . In particular, [N : M ] (+)N is a homogeneous ideal of R(M). The purpose of this paper is to investigate how properties of the ideal [N : M ](+)N are related to those of N . We determine when R(M) is a μ-ring, strongly Laskerin ring, Hilbert ring or satisfies Property (U) or Property (FU). It is also shown that if all homogeneous ideals of R(M) have a certain prescribed property, then all ideals of R(M) have the same property.