Relative Injective Modules and Relative Flat Modules

Authors: Jianmin Xing, Rufeng Xing

Abstract:

Let R be a ring, n a fixed nonnegative integer. The concepts of (n, 0)-FI-injective and (n, 0)-FI-flat modules, and then give some characterizations of these modules over left n-coherent rings are introduced . In addition, we investigate the left and right n-FI-resolutions of R-modules by left (right) derived functors Extn(−,−) (Torn(−,−) ) over a left n-coherent ring, where n-FI stands for the categories of all (n, 0)- injective left R-modules. These modules together with the left or right derived functors are used to study the (n, 0)-injective dimensions of modules and rings.

Keywords: (n, 0)-injective module, (n, 0)-injective dimension, (n, 0)-FI-injective(flat) module, (Pre)cover, (Pre)envelope.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1091096

References:

[1] J.L.Chen, N.Q.Ding, On n-coherent rings. Comm.Algebra 24, 1996, pp. 3211–3216.
[2] R.R.Colby, Rings which have flat injective modules, J.Algebra 35, 1975, pp.239–252.
[3] D.L.Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra.
[4] N.Q.Ding, J.L.Chen, Relative coherence and preenvelopes, Manuscripta Math., 81 , 1993, pp.243–262. London Math.Soc.85(2002) 393-440.
[5] E.E.Enochs, Injective and flat covers, envelopes and resolvents, Israel J.Math. 39, 1981, pp.189–209.
[6] E.E.Enochs, O.M.G. Jenda, Relative Homological Algebra, deGruyter Exp.Math.,30, deGruyter, Berlin, 2000.
[7] L.X.Mao.,N.Q.Ding ,FI-injective and FI-flat modules, J. Algebra 309, 2007, pp.367–385.
[8] L.X.Mao.,N.Q.Ding, Relative Projective module and relative injective module , Comm. Algebra 34 (2006),2403–2418.
[9] K.R.Pinzon, Absolutely pure modules, PhDThesis,University of Kentucky, 2005.
[10] J.J.Rotman, An Introduction to Homological Algebra, Academic Press, NewYork, 1979.
[11] B.Stenstrom, Coherent rings and FP-injective modules, J. London Math.Soc. 2, 1970, pp.323–329.
[12] J.Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag, Berlin, 1996.
[13] D.X.Zhou, On n-coherent rings and (n, d)-Rings, Comm.Algebra 32(6), 2004, pp.2425–2441.
[14] Z.M. Zhu, On n-Coherent Ring, n-hereditary ring and n-regular Ring Bulletin of the Iranian Mathematical Society 37(4), 2011, pp. 251-267.