Integral Dependence and the Nullstellensatz

Abstract

The problem of solving equations and saying something about the solutions is a fundamental motivation and goal of commutative algebra. In pursuing this goal, it is often important to adjoin a solution of a polynomial equation in one variable: Given a ring R and a polynomial p(x) ∈ R[x], the ring R[x]/(p) may be thought of as the result of adjoining a root of p to R as freely as possible; the root adjoined is of course the image of x.