A homological characterization of Q0-Prüfer v-multiplication rings

Yıl 2022, , 228 - 240, 16.07.2022

Xiaolei Zhang

https://doi.org/10.24330/ieja.1077664

Öz

Let $R$ be a commutative ring. An $R$-module $M$ is called a semi-regular $w$-flat module if $\Tor_1^R(R/I,M)$ is $\GV$-torsion for any finitely generated semi-regular ideal $I$. In this article, we show that the class of semi-regular $w$-flat modules is a covering class. Moreover, we introduce the semi-regular $w$-flat dimensions of $R$-modules and the $sr$-$w$-weak global dimensions of the commutative ring $R$. Utilizing these notions, we give some homological characterizations of $\WQ$-rings and $Q_0$-\PvMR s.

Anahtar Kelimeler

Semi-regular $w$-flat module, $sr$-$w$-weak global dimension, $\WQ$-ring, $Q_0$-\PvMR

Kaynakça

• D. D. Anderson, D. F. Anderson and R. Markanda, The rings $R(X)$and $R\langle X\rangle$, J. Algebra, 95(1) (1985), 96-115.
• L. Bican, R. E. Bashir and E. E. Enochs, All modules have flat covers, Bull. London Math. Soc., 33(4) (2001), 385-390.
• J. W. Brewer, D. L. Costa and K. McCrimmon, Seminormality and root closure in polynomial rings and algebraic curves, J. Algebra, 58(1) (1979), 217-226.
• H. S. Butts and W. Smith, Pr\"{u}fer rings, Math. Z., 95 (1967), 196-211.
• E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter, Berlin, 2000.
• L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, 84, American Mathematical Society, Providence, RI, 2001.
• S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
• H. Holm and P. Jorgensen, Covers, precovers, and purity, Illinois J. Math., 52(2) (2008), 691-703.
• J. A. Huckaba and I. J. Papick, Quotient rings of polynomial rings, Manuscripta Math., 31(1-3) (1980), 167-196.
• T. G. Lucas, Characterizing when $R[X]$ is integrally closed II, J. Pure Appl. Algebra, 61(1) (1989), 49-52.
• T. G. Lucas, Strong Prüfer rings and the ring of finite fractions, J. Pure Appl. Algebra, 84(1) (1993), 59-71.
• T. G. Lucas, Krull rings, Prüfer $v$-multiplication rings and the ring of finite fractions, Rocky Mountain J. Math., 35(4) (2005), 1251-1325.
• R. Matsuda, Notes on Prüfer $v$-multiplication ring, Bull. Fac. Sci. Ibaraki Univ., Math., 12 (1980), 9-15.
• H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1989.
• L. Qiao and F. G. Wang, $w$-Linked $Q_0$-overrings and $Q_0$-Prüfer $v$-Multiplication Rings, Comm. Algebra, 44(9) (2016), 4026-4040.
• F. G. Wang and H. Kim, Two generalizations of projective modules and their applications, J. Pure Appl. Algebra, 219(6) (2015), 2099-2123.
• F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016.
• F. G. Wang and L. Qiao, The $w$-weak global dimension of commutative rings, Bull. Korean Math. Soc., 52(4) (2015), 1327-1338.
• F. G. Wang, L. Qiao and D. C. Zhou, A homological characterization of strong Prüfer rings, Acta Math. Sinica (Chin. Ser.), 64(2) (2021), 311-316.
• F. G. Wang, D. C. Zhou and D. Chen, Module-theoretic characterizations of the ring of finite fractions of a commutative ring, J. Commut. Algebra, to appear. https://projecteuclid.org/euclid.jca/1589335712.
• F. G. Wang, D. C. Zhou, H. Kim, T. Xiong and X. W. Sun, Every Prüfer ring does not have small finitistic dimension at most one, Comm. Algebra, 48(12) (2020), 5311-5320.
• X. L. Zhang, Covering and enveloping on $w$-operation, J. Sichuan Normal Univ. (Nat. Sci.), 42(3) (2019), 382-386.
• X. L. Zhang, A homological characterizations of Prüfer $v$-multiplication rings, Bull. Korean Math. Soc., to appear.
• X. L. Zhang, G. C. Dai, X. L. Xiao and W. Qi, Semi-regular flat modules over strong Prüfer rings, https://arxiv.org/abs/2111.02221.
• X. L. Zhang and F. G. Wang, On characterizations of $w$-coherent rings II, Comm. Algebra, 49(9) (2021), 3926-3940.
• X. L. Zhang and F. G. Wang, The small finitistic dimensions over commutative rings, J. Commut. Algebra, to appear. https://arxiv.org/abs/2103.08807.
• D. C. Zhou, H. Kim, F. G. Wang and D. Chen, A new semistar operation on a commutative ring and its applications, Comm. Algebra, 48(9) (2020), 3973-3988.