Polynomial extensions of Jacobson rings

Watters, J. F.

doi:10.1016/0021-8693(75)90105-2

Cited by 27 publications

(13 citation statements)

References 3 publications

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“…Independent of the above, we offer in § 1 a short direct proof of the Jacobson ring result from [7]. Our purpose here is twofold.…”

Section: Introductionmentioning

confidence: 97%

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A note on Jacobson rings and polynomial rings

Ferrero¹,

Parmenter²

1989

Proc. Amer. Math. Soc.

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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,

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“…Independent of the above, we offer in § 1 a short direct proof of the Jacobson ring result from [7]. Our purpose here is twofold.…”

Section: Introductionmentioning

confidence: 97%

“…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.…”

Section: Introductionmentioning

confidence: 99%

A note on Jacobson rings and polynomial rings

Ferrero¹,

Parmenter²

1989

Proc. Amer. Math. Soc.

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“…A (noncommutative) ring R is named Jacobson ring by Watters in [12], if every prime ideal is the intersection of maximal left (or right) ideals. This is obviously equivalent to require that prime ideals are the intersection of primitive ideals.…”

Section: Introductionmentioning

confidence: 99%

“…This is obviously equivalent to require that prime ideals are the intersection of primitive ideals. As shown in Watters [12], R is a Jacobson ring if and only if the polynomial ring R[X] is a Jacobson ring.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Hilbert Rings

Kaučikas¹,

Wisbauer

2004

J. Algebra Appl.

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Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n ∈ N every maximal ideal M ⊂ R[X 1 , . . . , X n ] contracts to a maximal ideal of R. This definition is also equivalent to R[X 1 , . . . , X n ]/M being a finitely generated as an R/M ∩ R-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

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“…Verifications of the appropriate closure properties which are not straight-forward can be found in [1], [2], [3] for semilocal and semiprimary rings, in [6] for nilpotent rings, and in [11], [17] for Pi-rings. The class of Jacobson rings is a left and right hereditary radical class [9], [18] from which fact analogues of properties (ii), (iv), and (v) are immediate. The other properties of Jacobson rings are verified in [2], [4], [16].…”

Section: Iv) a Homomorphic Image Of A Perfect Ring Is Perfect (V) Lementioning

confidence: 99%