Multiplication Modules and Homogeneous Idealization IV


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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


Sultanate of Oman

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 IM\subseteq N. In particular, \left[  N:M\right]  (+)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.

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

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

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