Abstract
In this paper, new advances on the understanding the structure of p.p. rings and their generalizations have been made. Especially among them, it is proved that a commutative ring R is a generalized p.p. ring if and only if R is a generalized p.f. ring and its minimal spectrum is Zariski compact, or equivalently, \(R/\mathfrak {N}\) is a p.p. ring and \(R_{\mathfrak {m}}\) is a primary ring for all \(\mathfrak {m}\in {\text {Max}}(R)\). Some of the major results of the literature either are improved or are proven by new methods. In particular, we give a new and quite elementary proof to the fact that a commutative ring R is a p.p. ring if and only if R[x] is a p.p. ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aghajani, M., Tarizadeh, A.: Characterizations of Gelfand rings specially clean rings and their dual rings. Results Math. 75(3), 125 (2020)
Aghajani, M.: N-pure ideals and mid rings. arXiv:2011.02134 (2020)
Al-Ezeh, H.: Exchange pf-rings and almost pp-rings. Int. J. Math. Math. Sci. 12(4), 725–728 (1989)
Al-Ezeh, H.: On generalized pf-rings. Math. J. Okayama Univ. 31, 25–29 (1989)
Barthel, T., Schlank, T.M., Stapleton, N.: Chromatic homotopy theory is asymptotically algebraic. Invent. Math. 220, 737–845 (2020)
Becker, J., et al.: Ultraproducts and approximation in local rings I. Invent. Math. 51, 189–203 (1979)
Contessa, M.: Ultraproducts of pm-rings and mp-rings. J. Pure Appl. Algebra 32, 11–20 (1984)
Denef, J., Lipshitz, L.: Ultraproducts and approximation in local rings II. Math. Ann. 253, 1–28 (1980)
Endo, S.: Note on p.p. rings. Nagoya Math. J. 17, 167–170 (1960)
Erman, D., et al.: Big polynomial rings and Stillman’s conjecture. Invent. Math. 218(2), 413–439 (2019)
Glaz, S.: Commutative Coherent Rings. Lecture Notes in Mathematics, p. 1371. Springer, Berlin (1989)
Glaz, S.: Controlling the Zero Divisors of a Commutative Ring. Lecture Notes in Pure and Applied Mathematics, vol. 231, pp. 191–212. Marcel Dekker, New York (2002)
Hirano, Y.: On generalized p.p. rings. Math J. Okayama Univ. 25(1), 7–11 (1983)
Huckaba, J.A., Keller, J.M.: Annihilation of ideals in commutative rings. Pac. J. Math. 83(2), 375–379 (1979)
Huckaba, J.A.: Commutative Rings with Zero Divisors. Marcel Dekker Inc, New York (1988)
Huh, C., et al.: p.p. rings and generalized p.p. rings. J. Pure Appl. Algebra 167(1), 37–52 (2002)
Jondrup, S.: p.p. rings and finitely generated flat ideals. Proc. Am. Math. Soc. 28(2), 431–435 (1971)
Knox, M.L., et al.: Generalizations of complemented rings with applications to rings of functions. J. Algebra Appl. 8(1), 17–40 (2009)
Koşan, M.T., Lee, T.K., Zhou, Y.: Feebly Baer rings and modules. Commun. Algebra 42(10), 4281–4295 (2014)
Matlis, E.: The minimal prime spectrum of a reduced ring. Ill. J. Math. 27, 353–391 (1983)
Quentel, Y.: Sur la compacité du spectre minimal d’un anneau. Bull. Soc. Math. France 99, 265–272 (1971)
Schoutens, H.: The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics, vol. 1999. Springer, Berlin (2010)
Tarizadeh, A.: Some results on pure ideals and trace ideals of projective modules. Acta Math. Vietnam. (to appear). arXiv:2002.10139 (2021)
Tarizadeh, A.: Notes on finitely generated flat modules. Bull. Korean Math. Soc. 57(2), 419–427 (2020)
Tarizadeh, A., Aghajani, M.: Structural results on harmonic rings and lessened rings. Beitr. Algebra Geom. (2021). https://doi.org/10.1007/s13366-020-00556-x
Tarizadeh, A.: On flat epimorphisms of rings and pointwise localizations (to appear in Mathematica). arXiv:1608.05835 (2021)
Vasconcelos, W.: Finiteness in projective ideals. J. Algebra 25, 269–278 (1973)
Acknowledgements
The author would like to give sincere thanks to the referee for very careful reading of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tarizadeh, A. Structure theory of p.p. rings and their generalizations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 178 (2021). https://doi.org/10.1007/s13398-021-01120-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-021-01120-5