数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (6): 2085-2095.doi: 10.1016/S0252-9602(12)60161-1

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IFP-FLAT DIMENSIONS AND IFP-INJECTIVE DIMENSIONS

卢博|刘仲奎   

  1. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
  • 收稿日期:2011-08-27 修回日期:2012-05-16 出版日期:2012-11-20 发布日期:2012-11-20
  • 基金资助:

    This research was supported by National Natural Science Foundation of China (10961021, 11001222).

IFP-FLAT DIMENSIONS AND IFP-INJECTIVE DIMENSIONS

 LU Bo, LIU Zhong-Kui   

  1. Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
  • Received:2011-08-27 Revised:2012-05-16 Online:2012-11-20 Published:2012-11-20
  • Supported by:

    This research was supported by National Natural Science Foundation of China (10961021, 11001222).

摘要:

In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd(M) =IFP-id(M+) and IFP-fd(M+)=IFP-id(M) for any R-module M over any ring R. Let IIn (resp., IFn) be the class of all left (resp., right) R-modules of IFP-injective (resp., IFP-flat) dimension at most n. We prove that every right R-module has an IFn-preenvelope, (IFn, IFn ) is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id(RR) ≤n, (IIn, IIn ) is a perfect cotorsion theory. This generalizes and improves the earlier work (J. Algebra 242 (2001), 447-459). Finally, some applications are given.

关键词: IFP-flat dimension, IFP-injective dimension, Pre (Cover), Pre (Envelope), Cotorsion theory, IFP-cotorsion module

Abstract:

In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd(M) =IFP-id(M+) and IFP-fd(M+)=IFP-id(M) for any R-module M over any ring R. Let IIn (resp., IFn) be the class of all left (resp., right) R-modules of IFP-injective (resp., IFP-flat) dimension at most n. We prove that every right R-module has an IFn-preenvelope, (IFn, IFn ) is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id(RR) ≤n, (IIn, IIn ) is a perfect cotorsion theory. This generalizes and improves the earlier work (J. Algebra 242 (2001), 447-459). Finally, some applications are given.

Key words: IFP-flat dimension, IFP-injective dimension, Pre (Cover), Pre (Envelope), Cotorsion theory, IFP-cotorsion module

中图分类号: 

  • 16D50