Introduction.

Let (Q, m, k, d) be a local ring. In this section we are concerned with the definition of a general element x of an ideal I of Q or, more generally, given a set of ideals Il, …, ls of Q, of a set of independent general elements Xl, …, Xs of the ideals Il, …, Is. The elements x, x1, …, xS are not elements of Q, but of Qg or of QN for N large. To be precise, x belongs to IQg (orlQN) and xj to IjQg (or IjQN).

In the account that follows, for typographical reasons, we will often use alternative notation for certain symbols. We now make this more precise. The elements X1, X2, … of the countable sequence of indeterminates used in the definition of the ring Qg will occasionally be written as X(1), X(2), … Similarly, where we have a set of elements indexed by a set of suffixes I1, I2, …, Is, rather than involve the use of suffix upon suffix, we will use a notation such as u(I1, I2, … Is), but probably not u(i(1), …, i(s). Finally, a sequence of symbols such as r1, r2,…,rs may be represented by a single capital letter R, and we will then define Ri to mean the sequence obtained by omitting the 1th term of the sequence, i.e., the sequence r1, r2, …, ri-1, ri+1, …, rs.