Czechoslovak Mathematical Journal, first online, pp. 1-16
Projectively coresolved Gorenstein flat modules over trivial ring extensions
Zhanping Wang, Yuanhui Jin, Jianyuan He
Received November 15, 2023. Published online February 18, 2025.
Abstract: Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. Sufficient and necessary conditions are established for projectively coresolved Gorenstein flat (PGF, for short) modules over $R\ltimes M$. More pecisely, it is proved that $(X, \alpha)$ is a PGF left $R\ltimes M$-module if and only if ${\rm Coker}(\alpha)$ is a PGF left $R$-module and the sequence $M \otimes_RM \otimes_RX \overset{M \otimes\alpha} \to\longrightarrow M \otimes_RX \overset{\alpha} \to\longrightarrow X$ is exact under some assumptions on $M$. As applications, it is characterized PGF modules over Morita rings with zero bimodule homomorphisms.
Keywords: PGF module; trivial ring extension; Morita ring
References: [1] D. Asefa: Ding projective modules over Morita context rings. Commun. Algebra 52 (2024), 79-87. DOI 10.1080/00927872.2023.2233032 | MR 4685755 | Zbl 07790874
[2] M. Auslander, M. Bridger: Stable Module Theory. Memoirs of the American Mathematical Society 94. AMS, Providence (1969). DOI 10.1090/memo/0094 | MR 0269685 | Zbl 0204.36402
[3] H. Bass: The Morita Theorems. University of Oregon, Eugene (1969).
[4] J. Cui, S. Rong, P. Zhang: Cotorsion pairs and model structures on Morita rings. J. Algebra 661 (2025), 1-81. DOI 10.1016/j.jalgebra.2024.08.001 | MR 4788649 | Zbl 07938126
[5] T. Dumitrescu, N. Mahdou, Y. Zahir: Radical factorization for trivial extensions and amalgamated duplication rings. J. Algebra Appl. 20 (2021), Article ID 2150025, 10 pages. DOI 10.1142/S0219498821500250 | MR 4221776 | Zbl 1460.13006
[6] E. E. Enochs, M. Cortés-Izurdiaga, B. Torrecillas: Gorenstein conditions over triangular matrix rings. J. Pure Appl. Algebra 218 (2014), 1544-1554. DOI 10.1016/j.jpaa.2013.12.006 | MR 3175039 | Zbl 1312.16023
[7] E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. DOI 10.1007/BF02572634 | MR 1363858 | Zbl 0845.16005
[8] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. de Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[9] E. E. Enochs, O. M. G. Jenda, B. Torrecillas: Gorenstein flat modules. J. Nanjing Univ., Math. Biq. 10 (1993), 1-9. MR 1248299 | Zbl 0794.16001
[10] R. M. Fossum, P. A. Griffith, I. Reiten: Trivial Extensions of Abelian Categories: Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory. Lecture Notes in Mathematics 456. Springer, Berlin (1975). DOI 10.1007/BFb0065404 | MR 0389981 | Zbl 0303.18006
[11] N. Gao, J. Ma, X.-Y. Liu: RSS equivalences over a class of Morita rings. J. Algebra 573 (2021), 336-363. DOI 10.1016/j.jalgebra.2020.12.037 | MR 4203536 | Zbl 1471.16006
[12] E. L. Green: On the representation theory of rings in matrix form. Pac. J. Math. 100 (1982), 123-138. DOI 10.2140/pjm.1982.100.123 | MR 0661444 | Zbl 0502.16016
[13] Q. Guo, C. Xi: Gorenstein projective modules over rings of Morita contexts. Sci. China, Math. 67 (2024), 2453-2484. DOI 10.1007/s11425-022-2206-8 | MR 4803497 | Zbl 07936927
[14] H. Holm: Gorenstein Homological Algebra: Ph. D. Thesis. University of Copenhagen, Copenhagen (2004).
[15] H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. DOI 10.1016/j.jpaa.2003.11.007 | MR 2038564 | Zbl 1050.16003
[16] P. Krylov, A. Tuganbaev: Formal Matrices. Algebra and Applications 23. Springer, Cham (2017). DOI 10.1007/978-3-319-53907-2 | MR 3642603 | Zbl 1367.16001
[17] L. Mao: Cotorsion pairs and approximation classes over formal triangular matrix rings. J. Pure Appl. Algebra 224 (2020), Article ID 106271, 21 pages. DOI 10.1016/j.jpaa.2019.106271 | MR 4048527 | Zbl 1479.16003
[18] L. Mao: Gorenstein flat modules and dimensions over formal triangular matrix rings. J. Pure Appl. Algebra 224 (2020), Article ID 106207, 10 pages. DOI 10.1016/j.jpaa.2019.106207 | MR 4021921 | Zbl 1453.16001
[19] L. Mao: Homological dimensions of special modules over formal triangular matrix rings. J. Algebra Appl. 21 (2022), Article ID 2250146, 14 pages. DOI 10.1142/S0219498822501468 | MR 4448429 | Zbl 1514.16011
[20] L. Mao: Ding projective and Ding injective modules over trivial ring extensions. Czech. Math. J. 73 (2023), 903-919. DOI 10.21136/CMJ.2023.0351-22 | MR 4632864 | Zbl 07729544
[21] L. Mao: Gorenstein flat modules over some Morita context rings. J. Algebra Appl. 23 (2024), Article ID 2450096, 16 pages. DOI 10.1142/S0219498824500968 | MR 4692618 | Zbl 07797226
[22] L. Mao: Gorenstein projective, injective and flat modules over trivial ring extensions. J. Algebra Appl. 24 (2025), Article ID 2550030. DOI 10.1142/S0219498825500306 | MR 4834025 | Zbl 7960960
[23] H. Minamoto, K. Yamaura: Homological dimension formulas for trivial extension algebras. J. Pure Appl. Algebra 224 (2020), Article ID 106344, 30 pages. DOI 10.1016/j.jpaa.2020.106344 | MR 4074582 | Zbl 1436.16010
[24] K. Morita: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Diagaku Sect. A 6 (1958), 83-142. MR 0096700 | Zbl 0080.25702
[25] M. Nagata: Local Rings. Interscience Tracts in Pure and Applied Mathematics 13. John Wiley & Sons, New York (1962). MR 0155856 | Zbl 0123.03402
[26] I. Palmér, J.-E. Roos: Explicit formulae for the global homological dimensions of trivial extensions of rings. J. Algebra 27 (1973), 380-413. DOI 10.1016/0021-8693(73)90113-0 | MR 0376766 | Zbl 0269.16019
[27] I. Reiten: Trivial Extensions and Gorenstein Rings: Ph. D. Thesis. University of Illinois, Urbana (1971). MR 2621542
[28] J. J. Rotman: An Introduction to Homological Algebra. Pure and Applied Mathematics 85. Academic Press, New York (1979). MR 0538169 | Zbl 0441.18018
[29] J. Šaroch, J. Šťovíček: Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules. Sel. Math., New Ser. 26 (2020), Article ID 23, 40 pages. DOI 10.1007/s00029-020-0543-2 | MR 4076700 | Zbl 1444.16010
[30] P. Zhang: Gorenstein-projective modules and symmetric recollements. J. Algebra 388 (2013), 65-80. DOI 10.1016/j.jalgebra.2013.05.008 | MR 3061678 | Zbl 1351.16012
Affiliations: Zhanping Wang (corresponding author), Yuanhui Jin, Jianyuan He, Department of Mathematics, Shaoxing University, No. 900 Chengnan Avenue, Yuecheng District, Shaoxing 312000, Zhejiang, P. R. China, e-mail: wangzp@nwnu.edu.cn, 1448756808@qq.com, 1729267164@qq.com
