Journal of Commutative Algebra

When is a Nakayama closure semiprime?

Janet C. Vassilev

Full-text: Open access


Many well-known closure operations such as integral closure and tight closure are both semiprime operations and Nakayama closures. In this short note, we begin the study on the overlap between Nakayama closures and semiprime operations. We exhibit examples of closure operations which are either semiprime or Nakayama but not the other. In the case of a discrete valuation ring we show that a closure operation $c$ is Nakayama if and only if it is semiprime and \[ (0)^c=\bigcap_{n \geq 1} (I^n)^c \] for any ideal $I$.

Article information

J. Commut. Algebra, Volume 6, Number 3 (2014), 439-454.

First available in Project Euclid: 17 November 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A15: Ideals; multiplicative ideal theory 13C05: Structure, classification theorems

Closure operation semiprime op eration Nakayama closure


Vassilev, Janet C. When is a Nakayama closure semiprime?. J. Commut. Algebra 6 (2014), no. 3, 439--454. doi:10.1216/JCA-2014-6-3-439.

Export citation


  • N. Epstein, A tight closure analogue of analytic spread, Math. Proc. Camb. Phil. Soc. 139 (2005), 371–383.
  • ––––, Reductions and special parts of closures, J. Algebra 323 (2010), 2209-–2225.
  • ––––, A guide to closure operations in commutative algebra, Progr. Comm. Alg. II, Walter de Gruyter, Berlin, 2012.
  • W. Heinzer, L. Ratliff and D. Rush, Basically full ideals in local rings, J. Alg. 250 (2002), 371–396.
  • C. Huneke, Tight closure and its applications, CBMS Lect. Notes Math. 88, American Mathematical Society, Providence 1996.
  • D. Kirby, Closure operations on ideals and submodules, J. Lond. Math. Soc. 44 (1969), 283–291.
  • C. Polini, B. Ulrich and M. Vitulli, The core of zero-dimensional monomial ideals, Adv. Math. 211 (2007), 72-–93.
  • L. Ratliff and D. Rush, Asymptotic primes of delta closures of ideals, Comm. Alg. 30 (2002), 1513–1531.
  • ––––, Notes on ideal covers and associated primes, Pacific J. Math. 73 (1977), 169–191.
  • J. Vassilev, Structure on the set of closure operations of a commutative ring, J. Algebra 321 (2009), 2737–2753.
  • ––––, A look at the prime and semiprime operations of one-dimensional domains, Houst. J. Math. 38 (2012), 1-–15.