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.
