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Efficient generation of zero dimensional ideals in polynomial rings. (English) Zbl 0687.13011

We call an ideal I in a commutative noetherian ring A to be efficiently generated if the minimal number of generators of I is the same as that of \(I/I^ 2\). By a zero-dimensional ideal I in a ring A, we mean an ideal I of A such that the Krull dimension of the quotient ring A/I is zero.
In this paper we prove that any zero-dimensional ideal I in \(A=R[T_ 1,...,T_ n] \) \((R:\quad a\quad commutative\) noetherian ring) is efficiently generated if \(n\geq 2\) and in case of \(n=1\), I is efficiently generated provided \(I\cap R\) is also zero dimensional in R. - Moreover if R is a power series ring over a field or a regular spot over an infinite perfect field, we have shown that any zero dimensional ideal in R[T] is efficiently generated.
In the case when I is a maximal ideal of a polynomial ring, these results have been proved by E. D. Davis and A. V. Geramita [Trans. Am. Math. Soc. 231, 497-505 (1977; Zbl 0365.13008)] and S. M. Bhatwadekar [ibid. 270, 175-181 (1982; Zbl 0486.13006)], respectively.
Reviewer: P.L.N.Varma

MSC:

13E15 Commutative rings and modules of finite generation or presentation; number of generators
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F25 Formal power series rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
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References:

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