In this chapter we present the basic element construction which is the foundation of general position arguments in commutative ring theory. The version given here follows the one in Eisenbud and Evans (1973) rather closely.

As stated in Chapter 0 our main interest lies in the setting of local algebra. However, the nature of applications of basic element theory in the context of global algebra suggests a more general treatment. For this reason we will assume throughout the chapter only that R is a commutative Noetherian ring. We will be interested in the ideals of R in layers. The prime ideals of height zero will be the bottom layer. The next layer will consist of the prime ideals of height one, etc. The fundamental strategy is to arrange things in such a way that properties at height k force desired properties at height k + 1 for all but finitely many primes. Then exercising some care we are able to exhibit the properties in question for the remaining primes. Of course this is going on at all the levels simultaneously.

Let M be a finitely generated R-module, M′ a submodule of M and let P be a prime ideal of R. Then we say that M′ is w-fold basic in M at P provided the number of generators of (M/M′) is less than or equal to the number of generators of MP minus w.