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  • XIAO-WU CHEN (a1) and MING LU (a2)


Let $R$ be a two-sided Noetherian ring, and let $M$ be a nilpotent $R$ -bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_{R}(M)$ . Under certain conditions, the ring $R$ is Gorenstein if and only if so is $T_{R}(M)$ . We characterize Gorenstein projective $T_{R}(M)$ -modules in terms of $R$ -modules.



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[1] Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential , Ann. Inst. Fourier (Grenoble) 59(6) (2009), 25252590.
[2] Assem, I., Brustle, T. and Schiffler, R., Cluster-tilted algebras as trivial extensions , Bull. Lond. Math. Soc. 40 (2008), 151162.
[3] Auslander, M., Reiten, I. and Smalø, S. O., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36 , Cambridge University Press, 1995.
[4] Beligiannis, A., Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras , J. Algebra 288 (2005), 137211.
[5] Chen, X. W., Singularity categories, Schur functors and triangular matrix rings , Algebr. Represent. Theory 12 (2009), 181191.
[6] Chen, X. W., Three results on Frobenius categories , Math. Z. 270 (2012), 4358.
[7] Cohn, P. M., Algebra, Vol. 3, 2nd ed., John Wiley Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1991.
[8] Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra, De Gruyter Expositions in Math. 30 , Walter de Gruyter, Berlin, New York, 2000.
[9] Geiss, C., Leclerc, B. and Schroer, J., Quivers with relations for symmetrizable Cartan matrices I: Foundations , Invent. Math. 209 (2017), 61158.
[10] Geuenich, J., Quiver modulations and potentials, PhD Thesis, University of Bonn, 2017.
[11] Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc., Lecture Notes Ser. 119 , Cambridge University Press, 1988.
[12] Keller, B., Chain complexes and stable categories , Manuscripta Math. 67 (1990), 379417.
[13] Keller, B. and Reiten, I., Cluster-tilted algebras are Gorenstein and stably Calabi–Yau , Adv. Math. 211 (2007), 123151.
[14] Mac Lane, S., Categories for the Working Mathematician, Grad. Texts in Math. 5 , Springer-Verlag, New York, 1998.
[15] Li, F. and Ye, C., Representations of Frobenius-type triangular matrix algebras , Acta Math. Sin. (Engl. Ser.) 33(3) (2017), 341361.
[16] Luo, X. and Zhang, P., Monic representations and Gorenstein-projective modules , Pacific J. Math. 264(1) (2013), 163194.
[17] Minamoto, H., Ampleness of two-sided tilting complexes , Int. Math. Res. Not. IMRN 1 (2012), 67101.
[18] Minamoto, H. and Yamaura, K., Homological dimension formulas for trivial extension algebras, preprint, 2017, arXiv:1710.01469v1.
[19] Roganov, Yu. V., The dimension of the tensor algebra of a projective bimodule , Math. Notes 18(5–6) (1975), 11191123.
[20] Wang, R., Gorenstein triangular matrix rings and category algebras , J. Pure Appl. Algebra 220(2) (2016), 666682.
[21] Xiong, B. L. and Zhang, P., Gorenstein-projective modules over triangular matrix Artin algebras , J. Algebra Appl. 11(4) (2012), 1250066.
[22] Zaks, A., Injective dimensions of semiprimary rings , J. Algebra 13 (1969), 7389.
[23] Zhang, P., Gorenstein-projective modules and symmetric recollements , J. Algebra 388 (2013), 6580.
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  • XIAO-WU CHEN (a1) and MING LU (a2)


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