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Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), if and only if the R-modules \(H^i_I(R)\) are I-cofinite with dimension not exceeding 1, for all integers \(i\ge 2\); in addition, under these equivalent conditions it is shown that, for each minimal prime ideal \({{\mathfrak {p}}}\) over I, either \({{\text {height}}}{{\mathfrak {p}}}\le 1\) or \(\dim R/{{\mathfrak {p}}}\le 1\), and the prime spectrum of the I-transform R-algebra \(D_I(R)\) 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 R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), if and only if for each minimal prime ideal \({\mathfrak {P}}\) of \({\widehat{R}}\), either \(\dim {\widehat{R}}/(I{\widehat{R}}+{\mathfrak {P}})\le 1\) or \(H^i_{I{\widehat{R}}}({\widehat{R}}/{\mathfrak {P}})=0\), for all integers \(i\ge 2\). Finally, it is shown that if R is a semi-local ring and the R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), then the category of all I-cofinite modules forms an Abelian subcategory of the category of all R-modules.
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