Vol. 19, No. 2, 1966

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ISSN: 0030-8730
The pseudo-radical of a commutative ring

Robert William Gilmer, Jr.

Vol. 19 (1966), No. 2, 275–284
Abstract

If D is an integral domain with identity having quotient field K, the pseudo-radical of D is defined to be the intersection of all nonzero prime ideals of D. Consideration of the pseudo-radical arises naturally in examining the relation between the statements “D has Jacobson radical zero” and “D[u] has Jacobson radical zero, where u K”. Theorem 4 proves that the first statement implies the second. As a corollary it follows that if M is a prime ideal of the polynomial ring R[X] over a commutative ring R and if P = M R, then M is an intersection of maximal ideals of R[X] if P is an intersection of maximal ideals of R. Consequently, if R is a Hilbert ring, R[X] is also a Hilbert ring. The remainder of the paper is devoted to a study of domains having nonzero pseudo-radical.

Goldman has defined in [6] the concept of a Hilbert ring: the commutative ring R with identity is a Hilbert ring if each proper prime ideal of R is an intersection of maximal ideals; here proper means an ideal different from R. The terminology is motivated by the observation that Hilbert’s Nullstellensatz may be interpreted as asserting that each proper prime ideal of the polynomial domain K[X1,,Xn] for K a field, is an intersection of maximal ideals. In work done independently but at approximately the same time, Krull introduced in [10] the concept of a Jacobson ring; the reason for the terminology being obvious from the definition: the commutative ring R with identity is a Jacobson ring if for each proper ideal A of R, the radical of R∕A and the Jacobson radical of R∕A coincide. From these two definitions it is easily seen that R is a Hilbert ring if and only if R is a Jacobson ring. [10; p. 359]. In the remainder of this paper we shall use the term Hilbert ring for the notions described above.

Mathematical Subject Classification
Primary: 13.30
Secondary: 16.00
Milestones
Received: 23 August 1965
Published: 1 November 1966
Authors
Robert William Gilmer, Jr.