Multiplication Modules and Homogeneous Idealization
نویسنده
چکیده
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). MSC2000: 13C13, 13C05, 13A15
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