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A homological characterization of Q0-Prüfer v-multiplication rings

Year 2022, Volume: 32 Issue: 32, 228 - 240, 16.07.2022

Xiaolei Zhang

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

Abstract

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.

Keywords

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

References

  • 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.

Year 2022, Volume: 32 Issue: 32, 228 - 240, 16.07.2022

Xiaolei Zhang

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

Abstract

References

  • 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.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Xiaolei Zhang This is me

Publication Date July 16, 2022
Published in Issue Year 2022 Volume: 32 Issue: 32

Cite

APA Zhang, X. (2022). A homological characterization of Q0-Prüfer v-multiplication rings. International Electronic Journal of Algebra, 32(32), 228-240. https://doi.org/10.24330/ieja.1077664
AMA Zhang X. A homological characterization of Q0-Prüfer v-multiplication rings. IEJA. July 2022;32(32):228-240. doi:10.24330/ieja.1077664
Chicago Zhang, Xiaolei. “A Homological Characterization of Q0-Prüfer V-Multiplication Rings”. International Electronic Journal of Algebra 32, no. 32 (July 2022): 228-40. https://doi.org/10.24330/ieja.1077664.
EndNote Zhang X (July 1, 2022) A homological characterization of Q0-Prüfer v-multiplication rings. International Electronic Journal of Algebra 32 32 228–240.
IEEE X. Zhang, “A homological characterization of Q0-Prüfer v-multiplication rings”, IEJA, vol. 32, no. 32, pp. 228–240, 2022, doi: 10.24330/ieja.1077664.
ISNAD Zhang, Xiaolei. “A Homological Characterization of Q0-Prüfer V-Multiplication Rings”. International Electronic Journal of Algebra 32/32 (July 2022), 228-240. https://doi.org/10.24330/ieja.1077664.
JAMA Zhang X. A homological characterization of Q0-Prüfer v-multiplication rings. IEJA. 2022;32:228–240.
MLA Zhang, Xiaolei. “A Homological Characterization of Q0-Prüfer V-Multiplication Rings”. International Electronic Journal of Algebra, vol. 32, no. 32, 2022, pp. 228-40, doi:10.24330/ieja.1077664.
Vancouver Zhang X. A homological characterization of Q0-Prüfer v-multiplication rings. IEJA. 2022;32(32):228-40.