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

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 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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