A note on Jacobson rings and polynomial rings

Abstract

Abstract. As is well known, if R is a ring in which every prime ideal is an intersection of primitive ideals, the same is true of R [X]. The purpose of this paper is to give a general theorem which shows that the above result remains true when many other classes of prime ideals are considered in place of primitive ideals. IntroductionThroughout this paper we assume that R is a ring with identity element and R[X] is the polynomial ring over R in an indeterminate X. A ring R is said to be a Jacobson ring if every prime ideal of R is an intersection of primitive (either left or right) ideals. In [7], Waiters proved that if /? is a Jacobson ring, the polynomial ring R[X] is also a Jacobson ring. A similar result also holds for Brown-McCoy rings [8], i.e., rings in which every prime ideal is an intersection of maximal ideals.In this note, sé will always denote a class of prime rings. We say that an ideal P of R is an sé -ideal if R/P e sé . When every prime ideal of R is an intersection of sé -ideals, the ring R is said to be an sé-Jacobson ring. For example, if sé is the class of primitive (simple) rings, then an sé -Jacobson ring is a Jacobson (Brown-McCoy) ring.The main purpose of this paper is to prove the following Theorem 5. Assume that sé is a class of prime rings satisfying condition (A). If R is an sé-Jacobson ring, then so is R[X].Condition (A) is defined near the beginning of §2. Since primitive (simple) rings satisfy this condition, the above theorem includes as particular cases the results in [7 and 8]. However, we show that many other classes of prime rings satisfy condition (A) as well. Some examples include prime Noetherian rings,

When are Ore extensions over nil rings Brown--McCoy radicals?

“…The (σ, d)-pseudo radical P (σ,d) (R) of a ring R is defined as the intersection of all non-zero (σ, d)-prime ideals of R. The next lemma generalizes [3,Lemma 3], and its proof is similar. Proof.…”

Noncommutative Hilbert Rings

“…Note that the above observations on Jacobson and Brown-McCoy rings are subsumed in Theorem 5 of Ferrero-Parmenter [4].…”

On Jacobson Near-rings and Special Radicals

“…Much research in ring theory has been done on Jacobson and Brown-McCoy rings (see [1], [34], [38] and [39], and later [9], [10], [21], [22] and [23]). A ring R is said to be a Jacobson ring if every prime ideal of R is an intersection of primitive (either left or right) ideals.…”