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Primal ideals and isolated components in non-commutative rings Barnes, Wilfred Eaton

Abstract

In any associative ring R an element x is not right prime (nrp) to an ideal A if yRx ≤ A for some y [symbol omitted] A. An ideal is primal if the elements nrp to it form an ideal. These definitions differ from those of Curtis (American Journal of Mathematics, vol. 76 (1952), pp. 687-700) but reduce to them for rings with unit and A.C.C. for ideals. They also reduce to Fuchs’ (Proceedings of the American Mathematical Society, vol. 1 (1950), pp. l-8) for commutative rings. An ideal B is nrp to A if every element of B is nrp to A. Maximal nrp to A ideals always exist and their intersection is called the adjoint of A. In a class of rings, called uniform, the maximal nrp ideals of any ideal are prime. The A.C.C. implies uniformity, but not conversely. Results (similar to the classical Noether theory) on representations of an ideal as the intersection of primal ideals with prime adjoints are obtained which include those of Fuchs and Curtis. If A and B are ideals in any associative ring such that A ≤ B, the lower right isolated B-component of A, L(A,B), is the ideal sum of all ideals Am⁻¹ where m is right prime to B and Am⁻¹ = {x|xRm ≤ A}. The upper right isolated B-component of A, U(A,B), (which always contains L(A,B)) is the intersection of all ideals C≤ 2 A and such that every m right prime to B is right prime to C. If B is a maximal nrp to A ideal then L(A,B) and U(A,B) are called lower and upper right principal components of A. For B a prime ideal in a commutative ring, these definitions reduce to those of W. Krull. It is shown that every ideal in an associative ring is the intersection of its lower right principal components, and under certain conditions is also the intersection of its upper right principal components.

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