Generating ideals up to projective equivalence. (English) Zbl 0790.13001
Let \(I\) be an ideal of the commutative Noetherian ring \(R\). An ideal \(J \leq I\) is a reduction of \(I\) if \(JI^ p=I^{p+1}\) for some positive integer \(p\). There is a unique largest ideal reduced to \(I\), called the integral closure of \(I\). Two ideals \(I\) and \(J\) are projectively equivalent if some power of \(I\) and some (possibly other) power of \(J\) have the same integral closure.
The author proves the following theorem. If \(I\) is an ideal of a commutative Noetherian ring of dimension \(d\) then \(I\) is projectively equivalent to some ideal \(J\) of \(R\) such that \(J/J^ 2\) can be generated by \(d\) elements. This theorem is at the end of a long line of results, stretching back to Kronecker.
The author proves the following theorem. If \(I\) is an ideal of a commutative Noetherian ring of dimension \(d\) then \(I\) is projectively equivalent to some ideal \(J\) of \(R\) such that \(J/J^ 2\) can be generated by \(d\) elements. This theorem is at the end of a long line of results, stretching back to Kronecker.
Reviewer: B.A.F.Wehrfritz (London)
MSC:
13B22 | Integral closure of commutative rings and ideals |
13E15 | Commutative rings and modules of finite generation or presentation; number of generators |
13E05 | Commutative Noetherian rings and modules |
13A15 | Ideals and multiplicative ideal theory in commutative rings |
References:
[1] | M. Boratyński, Generating ideals up to radical, Arch. Math. (Basel) 33 (1979/80), no. 5, 423 – 425. · Zbl 0414.13004 · doi:10.1007/BF01222779 |
[2] | David Eisenbud and E. Graham Evans Jr., Every algebraic set in \?-space is the intersection of \? hypersurfaces, Invent. Math. 19 (1973), 107 – 112. · Zbl 0287.14002 · doi:10.1007/BF01418923 |
[3] | N. Mohan Kumar, On two conjectures about polynomial rings, Invent. Math. 46 (1978), no. 3, 225 – 236. · Zbl 0395.13009 · doi:10.1007/BF01390276 |
[4] | Ernst Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the German by Michael Ackerman; With a preface by David Mumford. · Zbl 0563.13001 |
[5] | Gennady Lyubeznik, A property of ideals in polynomial rings, Proc. Amer. Math. Soc. 98 (1986), no. 3, 399 – 400. · Zbl 0612.13008 |
[6] | Stephen McAdam, Finite coverings by ideals, Ring theory (Proc. Conf., Univ. Oklahoma, Norman, Okla., 1973) Dekker, New York, 1974, pp. 163 – 171. Lecture Notes in Pure and Appl. Math., Vol. 7. |
[7] | Stephen McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. · Zbl 0529.13001 |
[8] | D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145 – 158. · Zbl 0057.02601 |
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