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Paper IPM / M / 16301  


Abstract:  
Let I be an ideal of a commutative Noetherian ring R. It is shown that the Rmodules H^{i}_{I}(M) are Icofinite, for all finitely generated Rmodules M and all i ∈ \mathbb N_{0}, if and only if the Rmodules H^{i}_{I}(R) are Icofinite with dimension not exceeding 1, for all integers i ≥ 2; in addition, under these equivalent conditions it is shown that, for each minimal prime ideal \mathfrak p over I, either height\mathfrak p ≤ 1 or dimR/\mathfrak p ≤ 1, and the prime spectrum of the Itransform Ralgebra D_{I}(R) equipped with its Zariski topology is Noetherian. Also, by constructing an example we show that under the same equivalent conditions in general the ring D_{I}(R) need not be Noetherian. Furthermore, in the case that R is a local ring, it is shown that the Rmodules H^{i}_{I}(M) are Icofinite, for all finitely generated Rmodules M and all i ∈ \mathbb N_{0}, if and only if for each minimal prime ideal \mathfrak P of ∧R, either dim∧R/(I∧R+\mathfrak P) ≤ 1 or H^{i}_{I∧R}(∧R/\mathfrak P)=0, for all integers i ≥ 2. Finally, it is shown that if R is a semilocal ring and the Rmodules H^{i}_{I}(M) are Icofinite, for all finitely generated Rmodules M and all i ∈ \mathbb N_{0}, then the category of all Icofinite modules forms an Abelian subcategory of the category of all Rmodules.
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