Abstract
We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy prime avoidance. Examples and counterexamples are given throughout to illustrate the phenomena that can occur. As an application, we show how to use prime avoidance to construct counterexamples among rings essentially of finite type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Heinzer, W., Roitman, M.: Well-centered overrings of an integral domain. J. Algebra 272(2), 435–455 (2004)
Hochster, M., Huneke, C.: Localization and test exponents for tight closure. Michigan Math. J. 48(1), 305–329 (2000)
Pakala, J.V., Shores, T.S.: On compactly packed rings. Pac. J. Math. 97(1), 197–201 (1981)
Reis, C.M., Viswanathan, T.M.: A compactness property for prime ideals in Noetherian rings. Proc. Am. Math. Soc. 25, 353–356 (1970)
Sharp, R.Y., Vámos, P.: Baire’s category theorem and prime avoidance in complete local rings. Arch. Math. 44, 243–248 (1985)
Smith, W.W.: A covering condition for prime ideals. Proc. Am. Math. Soc. 30, 451–452 (1971)
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
Chen, J. Infinite prime avoidance. Beitr Algebra Geom 62, 587–593 (2021). https://doi.org/10.1007/s13366-020-00507-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-020-00507-6