Algebra & Number Theory

Factorially closed subrings of commutative rings

Sagnik Chakraborty, Rajendra Gurjar, and Masayoshi Miyanishi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove some new results about factorially closed subrings of commutative rings. We generalize this notion to quasifactorially closed subrings of commutative rings and prove some results about them from algebraic and geometric viewpoints. We show that quasifactorially closed subrings of polynomial and power series rings of dimension at most three are again polynomial (resp. power series) rings in a smaller number of variables. As an application of our results, we give a short proof of a result of Lê Dũng Tráng in connection with the Jacobian problem.

Article information

Source
Algebra Number Theory, Volume 9, Number 5 (2015), 1137-1158.

Dates
Received: 22 December 2014
Revised: 4 May 2015
Accepted: 9 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842358

Digital Object Identifier
doi:10.2140/ant.2015.9.1137

Mathematical Reviews number (MathSciNet)
MR3366001

Zentralblatt MATH identifier
1318.13002

Subjects
Primary: 13A05: Divisibility; factorizations [See also 13F15]
Secondary: 13B99: None of the above, but in this section 14R05: Classification of affine varieties

Keywords
factorially closed subring

Citation

Chakraborty, Sagnik; Gurjar, Rajendra; Miyanishi, Masayoshi. Factorially closed subrings of commutative rings. Algebra Number Theory 9 (2015), no. 5, 1137--1158. doi:10.2140/ant.2015.9.1137. https://projecteuclid.org/euclid.ant/1510842358


Export citation

References

  • M. Artin, “On the solutions of analytic equations”, Invent. Math. 5 (1968), 277–291.
  • E. Brieskorn, “Rationale Singularitäten komplexer Flächen”, Invent. Math. 4 (1968), 336–358.
  • N. Gupta, “On Zariski's cancellation problem in positive characteristic”, Adv. Math. 264 (2014), 296–307.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York-Heidelberg, 1977.
  • M. Miyanishi, “Normal affine subalgebras of a polynomial ring”, pp. 37–51 in Algebraic and topological theories (Kinosaki, 1984), edited by M. Nagata et al., Kinokuniya, Tokyo, 1986.
  • M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965.
  • W. D. Neumann and P. Norbury, “Nontrivial rational polynomials in two variables have reducible fibres”, Bull. Austral. Math. Soc. 58:3 (1998), 501–503.
  • P. Samuel, Lectures on unique factorization domains, TIFR 30, Tata Institute of Fundamental Research, Bombay, 1964.
  • L. D. Tráng, “Some remarks on relative monodromy”, pp. 397–403 in Real and complex singularities (Oslo, 1976), edited by P. Holm, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.
  • L. D. Tráng, “Simple rational polynomials and the Jacobian conjecture”, Publ. Res. Inst. Math. Sci. 44:2 (2008), 641–659.