Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ADMISSIBLE BALANCED PAIRS OVER FORMAL TRIANGULAR MATRIX RINGS

  • Mao, Lixin (Department of Mathematics and Physics Nanjing Institute of Technology)
  • Received : 2020.11.01
  • Accepted : 2021.08.19
  • Published : 2021.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Suppose that $T=\(\array{A&0\\U&B}\)$ is a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. Let ℭ1 and ℭ2 be two classes of left A-modules, 𝔇1 and 𝔇2 be two classes of left B-modules. We prove that (ℭ1, ℭ2) and (𝔇1, 𝔇2) are admissible balanced pairs if and only if (p(ℭ1, 𝔇1), h(ℭ2, 𝔇2) is an admissible balanced pair in T-Mod. Furthermore, we describe when ($P^{C_1}_{D_1}$, $I^{C_2}_{D_2}$) is an admissible balanced pair in T-Mod. As a consequence, we characterize when T is a left virtually Gorenstein ring.

Keywords

Acknowledgement

This work was financially supported by NSFC (11771202). The author wants to express his gratitude to the referee for the very helpful comments and suggestions.

References

  1. J. Asadollahi and S. Salarian, On the vanishing of Ext over formal triangular matrix rings, Forum Math. 18 (2006), no. 6, 951-966. https://doi.org/10.1515/FORUM.2006.048
  2. M. Auslander and S. O. Smalo, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61-122. https://doi.org/10.1016/0021-8693(80)90113-1
  3. A. Beligiannis and I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. https://doi.org/10.1090/memo/0883
  4. X. Chen, Homotopy equivalences induced by balanced pairs, J. Algebra 324 (2010), no. 10, 2718-2731. https://doi.org/10.1016/j.jalgebra.2010.09.002
  5. E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189-209. https://doi.org/10.1007/BF02760849
  6. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000. https://doi.org/10.1515/9783110803662
  7. S. Estrada, M. A. P'erez, and H. Zhu, Balanced pairs, cotorsion triplets and quiver representations, Proc. Edinb. Math. Soc. (2) 63 (2020), no. 1, 67-90. https://doi.org/10.1017/s0013091519000270
  8. R. M. Fossum, P. A. Griffith, and I. Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin, 1975.
  9. X. Fu, Y. Hu, and H. Yao, The resolution dimensions with respect to balanced pairs in the recollement of abelian categories, J. Korean Math. Soc. 56 (2019), no. 4, 1031-1048. https://doi.org/10.4134/JKMS.j180577
  10. R. Gobel and J. Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006. https://doi.org/10.1515/9783110199727
  11. E. L. Green, On the representation theory of rings in matrix form, Pacific J. Math. 100 (1982), no. 1, 123-138. http://projecteuclid.org/euclid.pjm/1102725383 https://doi.org/10.2140/pjm.1982.100.123
  12. A. Haghany and K. Varadarajan, Study of formal triangular matrix rings, Comm. Algebra 27 (1999), no. 11, 5507-5525. https://doi.org/10.1080/00927879908826770
  13. A. Haghany and K. Varadarajan, Study of modules over formal triangular matrix rings, J. Pure Appl. Algebra 147 (2000), no. 1, 41-58. https://doi.org/10.1016/S0022-4049(98)00129-7
  14. P. Krylov and A. Tuganbaev, Formal matrices, Algebra and Applications, 23, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-53907-2
  15. L. Mao, Cotorsion pairs and approximation classes over formal triangular matrix rings, J. Pure Appl. Algebra 224 (2020), no. 6, 106271, 21 pp. https://doi.org/10.1016/j.jpaa.2019.106271
  16. L. Mao, Gorenstein flat modules and dimensions over formal triangular matrix rings, J. Pure Appl. Algebra 224 (2020), no. 4, 106207, 10 pp. https://doi.org/10.1016/j.jpaa.2019.106207
  17. J. Saroch and J. Stovicek, Singular compactness and definability for Σ-cotorsion and Gorenstein modules, Selecta Math. (N.S.) 26 (2020), no. 2, Paper No. 23, 40 pp. https://doi.org/10.1007/s00029-020-0543-2
  18. J. Wang and L. Liang, A characterization of Gorenstein projective modules, Comm. Algebra 44 (2016), no. 4, 1420-1432. https://doi.org/10.1080/00927872.2015.1027356