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 |
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