Skew Polynimials and Jacobson Rings

Abstract

IntroductionIt was shown in [8] that if R is a Jacobson ring, that is a ring in which every prime ideal is an intersection of primitive ideals, and if a is an automorphism of R, then R[x, a] need not be a Jacobson ring. On the other hand, the results of [3] and [7] show that under suitable conditions skew polynomial extensions of Jacobson rings will be Jacobson rings. This paper arose out of an investigation into when R[x, a] is a Jacobson ring.The concept of the a-prime radical was introduced in [8] and here we define other radicals analogous to the Brown-McCoy, Jacobson, and nil radicals. The definition of an a-Jacobson ring and related ideas are formulated in § 1 and then in § 2 we show that R[x, a] is an a-Jacobson ring if and only if J? is an a-Jacobson ring. We believe this result to be the appropriate analogue of that obtained for polynomial rings over Jacobson rings in [9].Section 3 investigates further the a-radicals and we show how the theorem of Jordan and Jordan on the Jacobson radical of R[x, a], proved in [6], may be placed in the context of the a-nil radical.The paper ends with further discussion of the question which provoked the study, viz. when is R[x, a] a Jacobson ring? We obtain new sufficient conditions in Theorem 4.4 and then note that these conditions are satisfied by Noetherian rings, so giving the theorem of Goldie and Michler [3], and by the class of rings studied by King [7]. In Theorem 4:22 we obtain necessary and sufficient conditions for a large class of rings, including all commutative rings. P EA = {g G S: gx e M] and B = Sx. Since R is a-prime, we have A c M or i? <= i¥. In the latter case, M = Sx, which is impossible since P £ $£. Hence A^M and from MnR = 0 we get jfcTn£ I0] = 0. Now, by Lemma 2.12, M = P and so / = P, as required.2.14. THEOREM. Let R be an a-ring and S = R[x,a]. Then S is a-Jacobson if and only if R is a-Jacobson.Proof. The 'only if is clear since R = S/Sx and Sx is an a-invariant ideal of S.We prove the converse. Let P be an a-prime ideal of S. We need to prove that J a (P) = P. If x e P, this is clear as £/P = R/(P n i2). We assume that re ^ P. Since PnR is a-invariant we can factor out (PnR)S, which is contained in P, and assume that P n R = 0 and, b}' [8], that i? is a-prime and S is a-prime. In particular, P a (R) = 0 and therefore J a (R) = 0, since R is a-Jacobson. Hence by Corollary 2.6, J a (*S) = 0, and we are finished if P = 0.Assume now that P # 0. From Lemma 2.10, A(P)(J a {P) n R) = 0. Then, since i2 is an a-prime ring, J a (P)r\R = 0. It follows from Corollary 2.13 that J a (P) = P.

Radicals of skew polynomial rings and skew Laurent polynomial rings

“…Due to Lam, Leroy and Matcsuk [7], the upper σ -nil radical of R, denote by N σ (R), is given by N σ (R) = {I | I is σ -nil σ -ideal of R}. A subset S of R is called σ -n-nil (n 1) if every element in S is σ mnilpotent for any m n. Pearson, Stephenson and Watters [12] also introduced the σ -nil radical of R by the sum of all σ -invariant σ -n-nil ideals for some n 1. Note that the upper σ -nil radical is equal to the σ -nil radical by [7,Proposition 3.7].…”

“…where J σ (R) denotes the σ -Jacobson radical defined by the intersection of all the σ -primitive ideals of R [12,Proposition 3.9].…”

“…The concept of the upper σ -nil radical was also introduced in [7] which is the same concept as σ -nil radicals of [12] by [7,Proposition 3.7].…”

Characterizations of Elements in Prime Radicals of Skew Polynomial Rings and Skew Laurent Polynomial Rings

“…According to Pearson et al [5], an automorphism θ of R is called quasi-inner (QI for short) if there exists a regular element (i.e., neither left nor right zerodivisor) u ∈ R such that ur = θ(r)u for all r ∈ R, and θ is called power-quasiinner (PQI for short) if θ n is QI for some positive integer n.…”

Structure of Zero-Divisors in Skew Power Series Rings

“…Pearson and Stephenson [24] defined the σ-prime radical of a ring R, denoted by P σ (R), as the intersection of all strongly σ-prime ideals of R. Many authors have used the concept of σ-prime ideal to study of radicals of skew polynomial rings, see ( [7], [13], [19], [23], [24], [25], [26], etc.). A subset S of a ring R is called σ-nil if every element in S is σ-nilpotent.…”