Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A bracket power characterization
of analytic spread one ideals


Authors: L. J. Ratliff Jr. and D. E. Rush Jr.
Journal: Trans. Amer. Math. Soc. 352 (2000), 1647-1674
MSC (1991): Primary 13A15, 13B20, 13C10; Secondary 13H99
DOI: https://doi.org/10.1090/S0002-9947-99-02434-4
Published electronically: July 26, 1999
MathSciNet review: 1641107
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let $b_{1},\dots ,b_{g},x$ be regular nonunits in a local (Noetherian) ring $(R,M)$ and assume that $I$ $\subseteq $ $(xR)_{a}$, the integral closure of $xR$, where $I$ $=$ $(b_{1},\dots ,b_{g},x)R$. Then the main result shows that for all but finitely many units $u_{1},\dots ,u_{g}$ in $R$ that are non-congruent modulo $M$ and for all large integers $n$ and $k$ it holds that $I^{jn}$ $=$ $I^{[j]n}$ for $j$ $=$ $1,\dots ,k$ and $j$ not divisible by $char(R/M)$, where $I^{[j]}$ is the $j$-th bracket power $((b_{1}+u_{1}x)^{j}, \dots ,(b_{g}+u_{g}x)^{j},x^{j})R$ of $I$ $=$ $(b_{1}+u_{1}x, \dots ,b_{g}+u_{g}x,x)R$. And, conversely, if there exist positive integers $g$, $n$, and $k$ $\ge $ ${\binom{{n+g} }{{g}}}$ such that $I$ has a basis $\beta _{1},\dots ,\beta _{g} ,x$ such that $I^{kn}$ $=$ $({\beta _{1}}^{k},\dots ,{\beta _{g}}^{k},x^{k})^{n}R$, then $I$ has analytic spread one.


References [Enhancements On Off] (What's this?)

  • [ES] P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439-454. MR 54:7449
  • [HJLS] W. Heinzer, B. Johnston, D. Lantz, and K. Shah, The Ratliff-Rush ideal in a Noetherian ring: A survey, Methods In Module Theory, Lecture Notes in Pure and Applied Math, No. 140, 1993. MR 93k:13004
  • [KMOR] D. Katz, S. McAdam, J. Okon, and L. J. Ratliff, Jr., Essential prime divisors and projectively equivalent ideals, J. Algebra 109 (1987), 468-478. MR 88i:13016
  • [KR] D. Katz and L. J. Ratliff, Jr., U-essential prime divisors and sequences over an ideal, Nagoya Math. J. 103 (1986), 39-66. MR 87j:13002
  • [M] S. McAdam, Asymptotic Prime Divisors, LNM vol 1023, Springer-Verlag, 1983. MR 85f:13018
  • [MR] S. McAdam and L. J. Ratliff, Jr., Persistent primes and projective extensions of ideals, Comm. Algebra 16 (1988), 1141-1185. MR 89i:13003
  • [N] M. Nagata, Local Rings, Interscience Tracts In Pure and Applied Math. No. 13, Interscience, New York, NY, 1962. MR 27:5790
  • [NR] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Math. Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 15:59a
  • [RR1] L. J. Ratliff, Jr. and David E. Rush, Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), 929-934. MR 58:22034
  • [RR2] L. J. Ratliff, Jr. and David E. Rush, Triangular powers of integers from determinants of binomial coefficient matrices, Linear Algebra and Appl. (to appear).
  • [ZS1] O. Zariski and P. Samuel, Commutative Algebra, Vol. I, D. Van Nostrand Co., Inc., Princeton, NJ, 1958. MR 52:5641
  • [ZS2] O. Zariski and P. Samuel, Commutative Algebra, Vol. II, D. Van Nostrand Co., Inc., Princeton, NJ, 1960. MR 52:10706

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A15, 13B20, 13C10, 13H99

Retrieve articles in all journals with MSC (1991): 13A15, 13B20, 13C10, 13H99


Additional Information

L. J. Ratliff Jr.
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: ratliff@math.ucr.edu

D. E. Rush Jr.
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: rush@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02434-4
Keywords: Analytic spread, asymptotic prime divisor, binomial coefficient, bracket power of an ideal, essential prime divisor, integral closure of an ideal, local ring, Noetherian ring, persistent prime divisor, prenormal ideal, projectively equivalent ideals, Ratliff-Rush closure of an ideal, reduction of an ideal, superficial element
Received by editor(s): December 20, 1997
Published electronically: July 26, 1999
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society