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 Free Access

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 be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

**[ES]**Paul Eakin and Avinash Sathaye,*Prestable ideals*, J. Algebra**41**(1976), no. 2, 439–454. MR**0419428**, https://doi.org/10.1016/0021-8693(76)90192-7**[HJLS]**William Heinzer, Bernard Johnston, David Lantz, and Kishor Shah,*The Ratliff-Rush ideals in a Noetherian ring: a survey*, Methods in module theory (Colorado Springs, CO, 1991) Lecture Notes in Pure and Appl. Math., vol. 140, Dekker, New York, 1993, pp. 149–159. MR**1203805****[KMOR]**D. Katz, S. McAdam, J. S. Okon, and L. J. Ratliff Jr.,*Essential prime divisors and projectively equivalent ideals*, J. Algebra**109**(1987), no. 2, 468–478. MR**902964**, https://doi.org/10.1016/0021-8693(87)90151-7**[KR]**Daniel Katz and Louis J. Ratliff Jr.,*𝑢-essential prime divisors and sequences over an ideal*, Nagoya Math. J.**103**(1986), 39–66. MR**858471****[M]**Stephen McAdam,*Asymptotic prime divisors*, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. MR**722609****[MR]**S. McAdam and L. J. Ratliff Jr.,*Persistent primes and projective extensions of ideals*, Comm. Algebra**16**(1988), no. 6, 1141–1185. MR**939036**, https://doi.org/10.1080/00927878808823624**[N]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856****[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), no. 6, 929–934. MR**0506202**, https://doi.org/10.1512/iumj.1978.27.27062**[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]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. 1*, Springer-Verlag, New York-Heidelberg-Berlin, 1975. With the cooperation of I. S. Cohen; Corrected reprinting of the 1958 edition; Graduate Texts in Mathematics, No. 28. MR**0384768****[ZS2]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition; Graduate Texts in Mathematics, Vol. 29. MR**0389876**

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