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Cominimaxness with respect to ideals of dimension one. (English) Zbl 1359.13017
Summary: Let $$R$$ denote a commutative Noetherian (not necessarily local) ring and let $$I$$ be an ideal of $$R$$ of dimension one. The main purpose of this note is to show that the category $$\mathcal{M}(R,I)_{com}$$ of $$I$$-cominimax $$R$$-modules forms an abelian subcategory of the category of all $$R$$-modules. This assertion is a generalization of the main result of L. Melkersson [J. Algebra 372, 459–462 (2012; Zbl 1273.13029)]. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category $$\mathcal{C}_{B}^1(R)$$ of all $$R$$-modules of dimension at most one with finite Bass numbers forms an abelian subcategory of the category of all $$R$$-modules.

##### MSC:
 13D45 Local cohomology and commutative rings 14B15 Local cohomology and algebraic geometry 13E05 Commutative Noetherian rings and modules
##### Keywords:
arithmetic rank; Bass number; cominimax modules; minimax modules
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