Maximum number of independent elements and dimension of prime divisors in completions of local rings (Q1078253)

For an ideal \({\mathfrak a}\) of a commutative ring R with identity, elements \(a_ 1,...,a_ r\) of R are \({\mathfrak a}\)-independent if every form \(f(x_ 1,...,x_ r)\in R[x_ 1,...,x_ r]\) vanishes at \((a_ 1,...,a_ r)\) then f has coefficients all in \({\mathfrak a}\). This notion was introduced by \textit{G. Valla} [Rend. Sem. Mat. Univ. Padova 44(1970), 339-354 (1971; Zbl 0256.13010)] and has been studied by several authors. The main result in this article are \((1)\quad to\) give a good formula for \(\sup_{{\mathfrak b}}{\mathfrak a}\) which is defined to be the maximum of the numbers of \({\mathfrak a}\)-independent elements in an ideal \({\mathfrak b}\), in the case where the ring is noetherian and \({\mathfrak b}\subseteq \sqrt{{\mathfrak a}}\), and \((2)\quad to\) show some interesting applications of the formula. In particular, in {\S} 5 and {\S} 6, the author gives some fine improvements of works of \textit{W. Bruns} [J. Lond. Math. Soc., II. Ser. 22, 57-62 (1980; Zbl 0452.13005)] and \textit{L. J. Ratliff jun.} [Pac. J. Math. 91, 445-456 (1980; Zbl 0414.13002); J. Alg. 73, 327-343 (1981; Zbl 0471.13001) and 78, 410-430 (1982; Zbl 0498.13005)]. Some part of the article was announced by \textit{J.-E. Björk} in Sémin. d'algèbre, Dubreil-Malliavin, 34ème Année, Proc., Paris 1981, Lect. Notes Math. 924, 413-422 (1982; Zbl 0488.13002).