DP-PROJECTIVE MODULES AND DIMENSIONS

Year 2014, Volume: 15 Issue: 15, 101 - 116, 01.06.2014

Tiwei Zhao

https://doi.org/10.24330/ieja.266241

Cited By: 2

Abstract

In this paper, we introduce the notion of DP-projective modules.
It is shown that a left R-module M over a ring R is DP-projective if and only if
it is a cokernel of a Ding projective preenvelope f : A → B with B projective.
It is also shown that a ring R is semisimple if and only if every module is
DP-projective. Moreover, we investigate (global) DP-projective dimensions of
modules and rings. It is shown that l.DP-dim(R) = l.DP-ID(R). In addition,
other applications of those dimensions defined in this way are presented.

Keywords

DP-projective module, semisimple ring, DP-projective dimension

References

• F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed. Graduate Texts in Mathematics, 13, New York: Springer-Verlag, 1992.
• H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466–488.
• S. Breaz, Modules M such that ExtR(M, −) commuts with direct limits, Algebra Represent. Theory, (to appear), doi: 10.1007/s10468-012-9382-y.
• N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math., 78 (1993), 165–177.
• N. Q. Ding, Y. L. Li and L. X. Mao, Strongly Gorenstein flat modules, J. Aust. Math. Soc., 86 (2009), 323–338.
• E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin, 2000.
• X. H. Fu, H. Y. Zhu and N. Q. Ding, On copure projective modules and copure projective dimensions, Comm. Algebra, 40 (2012), 343–359.
• J. Gillespie, Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl., 12(1) (2010), 61–73.
• T. Ishikawa, On injective modules and flat modules, J. Math. Soc. Japan, 17(3) (1965), 291–296.
• C. U. Jensen, On the vanishing of lim(i), J. Algebra, 15(2) (1970), 151–166. ←−
• L. X. Mao and N. Q. Ding, Gorenstein F P -injective and Gorenstein flat mod- ules, J. Algebra Appl., 7(4) (2008), 491–506.
• N. Mahdou and M. Tamekkante, Strongly Gorenstein flat modules and dimen- sions, Chin. Ann. Math., 32B(4) (2011), 533–548.
• J. J. Rotman, An Introduction to Homological Algebra, 2nd ed. Springer, 2009.
• C. H. Yang, Strongly Gorenstein flat and Gorenstein FP-injective modules, Turk. J. Math., 37 (2013), 218–230.
• G. Yang, Homological properties of modules over Ding-Chen rings, J. Korean Math. Soc., 49 (1) (2012), 31–47.
• G. Yang, Z. K. Liu and L. Liang, Ding projective and Ding injective modules, Algebra Colloq., 20(4) (2013), 601-612.
• C. X. Zhang and L. M. Wang, Strongly Gorenstein flat dimensions, J. Math. Research & Exposition, 31(6) (2011), 977–988. Tiwei Zhao
• School of Mathematics and Computer Science Hubei University Wuhan, Hubei, People’s Republic of China e-mail: [e-mail]