ABSTRACT
Let Max(.A) denote the set of maximal prime divisors of A. For P <E Max(A), Krull defines the ideal A(p) = {x e R : xy £ A for some y e -R \ P} to be the principal component of A with respect to P and establishes the decomposition of every proper ideal A of R as the intersection of its principal components A — DpeMaxM) ^-(.P) [14, Satz 2]. However, as we discuss in [4], a drawback to this decomposition is that we do not know what kind of ideals the ^.(p) are. For instance, there sometimes exist elements of P that are prime to the principal component A(p); indeed, the ideal A(P) is not in general a primal ideal, where an ideal B is said to be primal if the set S(B) of elements non-prime to B is an ideal. If B is primal, then S(B) is a prime ideal called the adjoint prime of B.
