On maximal ideals in polynomial and Laurent polynomial rings (Q1188163)

An ideal \(I\) in a commutative noetherian ring \(A\) is said to be efficiently generated if its minimal number of generators coincides with the minimal number \(\mu\) of generators of the \(A\)-module \(I/I^ 2\). It is said to be projectively generated if there is a surjection of a projective \(A\)-module of rank \(\mu\) onto \(I\). Bhatwadekar has shown that any maximal ideal in the polynomial ring \(R[X]\) is projectively generated provided \(R\) is a commutative noetherian ring containing the field of rationals [see \textit{S. M. Bhatwadekar}, J. Algebra 91, 75-81 (1984; Zbl 0562.13017)]. The author proves the same for the Laurent polynomial ring \(R[X,X^{-1}]\) in case \(R\) is a complete local ring containing the rationals. Furthermore he shows that if \(R\) is a (not necessarily local) regular ring containing the rationals, then a maximal ideal in \(R[X]\) with height greater or equal to the dimension of \(R\) is efficiently generated. This result has to be viewed in connection with a result of \textit{M. Boratyński}, \textit{E. D. Davis} and \textit{A. V. Geramita} who proved that any maximal ideal in \(R[X]\) is efficiently generated if \(R\) is a regular ring of dimension at most 2 [see J. Algebra 51, 320-325 (1978; Zbl 0374.13007)].