×

Rings over which all modules are strongly Gorenstein projective. (English) Zbl 1194.13008

Gorenstein projective and Gorenstein injective modules were introduced by E. E. Enochs and O. M. G. Jenda [Math. Z. 220, No. 4, 611–633 (1995; Zbl 0845.16005)].
The main result of this paper shows, in particular, that for a commutative ring \(R\) the following conditions are equivalent: (i) Every \(R\)-module is Gorenstein projective; (ii) Every \(R\)-module is Gorenstein injective; (iii) Every Gorenstein projective \(R\)-module is Gorenstein injective; (iv) Every Gorenstein injective \(R\)-module is Gorenstein projective; (v) \(R\) is quasi-Frobenius. Hence, a \(G\)-semisimple ring is the same as a quasi-Frobenius ring.

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings

Citations:

Zbl 0845.16005
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] F.W. Anderson and K.R. Fuller, Rings and categories of modules , Grad. Texts Math. 13 , 2 -1992. · Zbl 0765.16001
[2] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings , Trans. Amer. Math. Soc. 95 (1960), 466-488. JSTOR: · Zbl 0094.02201 · doi:10.2307/1993568
[3] D. Bennis and N. Mahdou, Strongly Gorenstein projective, injective, and flat modules , J. Pure Appl. Algebra 210 (2007), 437-445. · Zbl 1118.13014 · doi:10.1016/j.jpaa.2006.10.010
[4] A.J. Berrick and M.E. Keating, An introduction to rings and modules , Cambridge University Press, Cambridge, 2000. · Zbl 0949.16001
[5] L.W. Christensen, Gorenstein dimensions , Lecture Notes Math. 1747 , Springer-Verlag, Berlin, 2000. · Zbl 0965.13010
[6] E.E. Enochs and O.M.G. Jenda, Relative homological algebra , de Gruyter Expositions Math. 30 , Walter de Gruyter & Co., Berlin, 2000. · Zbl 0952.13001
[7] H. Holm, Gorenstein homological dimensions , J. Pure Appl. Algebra 189 (2004), 167-193. · Zbl 1050.16003 · doi:10.1016/j.jpaa.2003.11.007
[8] W.K. Nicholson and M.F. Youssif, Quasi-Frobenius rings , Cambridge University Press, 158 , Cambridge, 2003. · Zbl 1042.16009
[9] J.J. Rotman, An introduction to homological algebra , Academic Press, New York, 1979. · Zbl 0441.18018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.