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Form rings and projective equivalence

Published online by Cambridge University Press:  24 October 2008

Daniel Katz  and
L. J. Ratliff Jr
Daniel Katz
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A.
L. J. Ratliff Jr
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, U.S.A.
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If I and J are ideals in a Noetherian ring R, then I and J are projectively equivalent in case (Ii)a = (Jj)a for some positive integers i, j (where Ka denotes the integral closure in R of the ideal K) and the form ring F(R, I) of R with respect to I is the graded ring R/I ⊕ I/I2 ⊕ I2/I3 ⊕ …. These two concepts have played an important role in many research problems in commutative algebra, so they have been deeply studied and many of their properties have been discovered. In a recent paper [13] they were combined to show that a semi-local ring R is unmixed if and only if for every ideal J in R there exists a projectively equivalent ideal J in R such that every prime divisor of zero in F(R, J) has the same depth. It seems to us that results similar to this are interesting and potentially quite useful, so in this paper we prove several additional such theorems. Namely, it is shown that all ideals in all local rings have a projectively equivalent ideal whose form ring is fairly nice. Also, a characterization similar to the just mentioned result in [13] is given for the class of local rings whose completions have no embedded prime divisors of zero, and several analogous new characterizations are given for locally unmixed Noetherian rings. In particular, it is shown that if I is an ideal in an unmixed local ring R such that height(I) = l(I) (where l(I) denotes the analytic spread of I), then there exists a projectively equivalent ideal J in R such that Ass (F(R, J)) has exactly m elements, all minimal, where m is the number of minimal prime divisors of I (so if I is open, then F(R, J) has exactly one prime divisor of zero and is a locally unmixed Noetherian ring).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 447 - 456
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Katz, D. and Ratliff, L. J. Jr. U-essential prime divisors and sequences over an ideal. Nagoya Math. J. (to appear).Google Scholar
[2]Katz, D. and Ratliff, L. J. Jr. Two notes on ideal-transforms (preprint).Google Scholar
[3]Mcadam, S.. Asymptotic Prime Divisors, Lecture Notes in Math. No. 1023 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[4]Mcadam, S. and Ratliff, L. J. Jr. Essential sequences. J. Algebra (to appear).Google Scholar
[5]Nagata, M.. On the chain problem of prime ideals. Nagoya Math. J. 10 (1956), 51–64.CrossRefGoogle Scholar
[6]Nagata, M.. Local Rings. Interscience Tracts 13 (Interscience, 1962).Google Scholar
[7]Okon, J. S. and Ratliff, L. J. Jr. Notes on analytic spread and certain sequences. Houston J. Math. (to appear).Google Scholar
[8]Ratliff, L. J. Jr. Locally quasi-unmized Noetherian rings and ideals of the principal class. Pacific J. Math. 52 (1974), 185–205.CrossRefGoogle Scholar
[9]Ratliff, L. J. Jr. On the prime divisors of zero in form rings. Pacific J. Math. 70 (1977), 489–517.Google Scholar
[10]Ratliff, L. J. Jr. Asymptotic sequences. J. Algebra 85 (1983), 337–360.CrossRefGoogle Scholar
[11]Ratliff, L. J. Jr. On linearly equivalent ideal topologies. J. Pure Appl. Algebra (to appear).Google Scholar
[12]Ratliff, L. J. Jr. Five notes on asymptotic prime divisors. Math. Z. (to appear).Google Scholar
[13]Ratliff, L. J. Jr. Three theorems on form rings. (Preprint).Google Scholar
[14]Rees, D.. A note on form rings and ideals. Mathematika 4 (1957), 51–60.CrossRefGoogle Scholar
[15]Zariski, O. and Samuel, P.. Commutative Algebra, Vol. II (Van Nostrand, 1960).CrossRefGoogle Scholar