Generating ideals in local rings using elements of high degrees (Q1116983)

Let (R,M) be a local Noetherian ring and I a nonzero ideal of R which can be minimally generated by r elements. A nonzero element x of R is said to have degree \( k\), written \(\deg (x)=k\), if \(x\in M^ k-M^{k+1}\). Suppose that \(x_ 1,...,x_ r\) generate I. What can be said about \(\deg (x_ 1)+...+\deg (x_ r) ?\) Certainly, if n is the least degree of any element of I, then \(\deg (x_ 1)+...+\deg (x_ r)\geq nr,\) and a basis \(x_ 1,...,x_ r\) for I can be chosen where equality holds. On the other hand, since \(I\cap M^ k\subseteq IM\) for large k, there is an upper bound for the sum \(\deg (x_ 1)+...+\deg (x_ r).\) The purpose of this paper is to give a simple construction of a basis \(x_ 1,...,x_ r\) for I with \(\deg (x_ 1)+...+\deg (x_ r)\) maximal and to show that this construction is essentially unique.