Primal Ideals and Isolated Components

Introduction. L. Fuchs [2 ] has given for Noetherian rings a theory of the representation of an ideal as an intersection of primal ideals, the theory being in many ways analogous to the classical Noether theory. An ideal Q is primal if the elements not prime to Q form an ideal, necessarily prime, called the adjoint of Q. Primary ideals are necessarily primal, but not conversely. Analogous results have been obtained by Curtis [l] for noncommutative rings with unit element, using a definition of right primal ideal which does not, however, reduce to that of Fuchs in a commutative ring. In this paper an alternative definition of right primal ideal in a general ring is given, which reduces to Fuchs' for commutative rings and to Curtis' for rings with unit element and ascending chain condition (A.C.C.) for ideals. This definition is based on a definition of "not right prime to A" which associates with any ideal A certain maximal not right prime to A ideals, analogous to Krull's maximal associated primes. These maximal not right prime to A ideals apparently are not necessarily prime unless a condition of "uniformity," which is weaker than the A.C.C, is imposed. In §3 a discussion of primal decompositions in rings without finiteness conditions is given, and in §4 the FuchsCurtis decomposition theorems are obtained for rings with A.C.C. for ideals. In §5 a new definition of the right associated primes of an ideal is given, and the maximal such ideals are determined. Following the methods of Murdoch [6], upper and lower right isolated 5-components of an ideal A, where B is any divisor of A, are defined and their properties investigated in §6. If B is a maximal not right prime to A ideal, the isolated .B-components of A are called upper and lower principal component ideals of A, and reduce to Krull's principal component ideals in commutative rings. It is shown in §7 that under certain conditions an ideal is the intersection of its upper principal components, and that any ideal in an associative ring is the intersection of its lower principal components. 1. Notation and definitions. We shall use R to denote an associative ring which will be noncommutative unless otherwise specified. The term ideal will always mean two-sided ideal. Proper ideals in R will be denoted by A, B, ■ • •