Efficient generation of zero dimensional ideals in polynomial rings
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 \textit{E. D. Davis} and \textit{A. V. Geramita} [Trans. Am. Math. Soc. 231, 497-505 (1977; Zbl 0365.13008)] and \textit{S. M. Bhatwadekar} [ibid. 270, 175-181 (1982; Zbl 0486.13006)], respectively.
scientific article; zbMATH DE number 4125522