# Multiplication Modules and Homogeneous Idealization IV

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 42, (2012), **Pages** 131-147

**Majid M. Ali **

Department of Mathematics and Statistics

Sultan Qaboos University

P.O. Box 36, PC. 123 Alkhoud

Muscat,

Sultanate of Oman mailto:mali@squ.edu.om

**Abstract** 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 . In particular,
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.

**Keywords** Multiplication module, Comultiplication module, Projective module, Flat module, μ-ring, Laskerian ring.

**Classification** (MSC2000) 13C13, 13C05, 13A15.