[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b160100

COMINIMAXNESS WITH RESPECT TO IDEALS OF DIMENSION ONE

Irani, Yavar (Department of Mathematics Islamic Azad University Meshkin-Shahr Branch)

Publication Information

Bulletin of the Korean Mathematical Society / v.54, no.1, 2017 , pp. 289-298 More about this Journal

Abstract

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 ${\mathfrak{M}}(R,\;I)_{com}$ 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 Melkersson in [15]. 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 ${\mathcal{C}}^1_B(R)$ of all R-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all R-modules.

Keywords

arithmetic rank; Bass number; cominimax modules; minimax modules;

Citations & Related Records

Reference

1	N. Abazari and K. Bahmanpour, Extension functors of local cohomology modules and Serre categories of modules, Taiwanese J. Math. 19 (2015), no. 1, 211-220. DOI
2	J. Azami, R. Naghipour, and B. Vakili, Finiteness properties of local cohomology modules for a-minimax modules, Proc. Amer. Math. Soc. 137 (2009), no. 2, 439-448. DOI
3	K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2359-2363. DOI
4	K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra. 321 (2009), no. 7, 1997-2011. DOI
5	K. Bahmanpour, R. Naghipour, and M. Sedghi, On the category of cofinite modules which is Abelian, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1101-1107. DOI
6	K. Bahmanpour, R. Naghipour, and M. Sedghi, On the finiteness of Bass numbers of local cohomology modules and cominimaxness, Houston J. Math. 40 (2014), no. 2, 319-337.
7	M. P. Brodmann and R. Y. Sharp, Local Cohomology; an algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998.
8	D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), no. 1, 45-52. DOI
9	R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164. DOI
10	K.-I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3275-3279. DOI
11	K.-I. Kawasaki, On a category of cofinite modules which is Abelian, Math. Z. 269 (2011), no. 1-2, 587-608. DOI
12	H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, UK, 1986.
13	L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Camb. Phil. Soc. 107 (1990), no. 2, 267-271. DOI
14	L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649-668. DOI
15	L. Melkersson, Cofiniteness with respect to ideals of dimension one, J. Algebra 372 (2012), 459-462. DOI
16	W. Vasconcelos, Divisor Theory in Module Categories, North-Holland, Amsterdam, 1974.
17	H. Zoschinger, Minimax moduln, J. Algebra 102 (1986), no. 1, 1-32. DOI
18	H. Zoschinger, Uber die maximalbedingung fur radikalvolle untermoduln, Hokkaido Math. J. 17 (1988), no. 1, 101-116. DOI