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INVARIANT POLYNOMIALS of ORE EXTENSIONS by Q-SKEW DERIVATIONS

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INVARIANT POLYNOMIALS of ORE EXTENSIONS by Q-SKEW DERIVATIONS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 11, November 2012, Pages 3739–3747 S 0002-9939(2012)11268-7 Article electronically published on March 12, 2012 INVARIANT POLYNOMIALS OF ORE EXTENSIONS BY q-SKEW DERIVATIONS CHEN-LIAN CHUANG, TSIU-KWEN LEE, AND CHENG-KAI LIU (Communicated by Harm Derksen) Abstract. Let R be a prime ring with the symmetric Martindale quotient ring Q. Suppose that δ is a quasi-algebraic q-skew σ-derivation of R.Fora minimal monic semi-invariant polynomial π(t)ofQ[t; σ, δ], we show that π(t) is also invariant if char R = 0 and that either π(t) − c for some c ∈ Q or π(t)p is a minimal monic invariant polynomial if char R = p ≥ 2. As an application, we prove that any R-disjoint prime ideal of R[t; σ, δ] is the principal ideal p(t) for an irreducible monic invariant polynomial p(t) unless σ or δ is X-inner. 1. Introduction It will be assumed throughout that R is an associative prime ring in the sense that for any a, b ∈ R, aRb = 0 implies a =0orb =0.Letσ be an automorphism of R.Byaσ-derivation of R,wemeanamapδ : R → R satisfying δ(x + y)=δ(x)+δ(y)andδ(xy)=δ(x)y + σ(x)δ(y) for all x, y ∈ R.Givenb ∈ R,themapadb,σ : x ∈ R → bx − σ(x)b defines a σ-derivation of R, called the inner σ-derivation defined by b. Analogously, for a −1 unit u ∈ R,themapIu : x ∈ R → uxu defines an automorphism of R, called the inner automorphism defined by the unit element u. Let Q be the symmetric Martindale quotient ring of R. The center of Q, denoted by C, is called the extended centroid of R. (See [1, Chapter 2] for details.) The σ-derivation δ of R, together with its automorphism σ, can be uniquely extended to a σ-derivation of Q [15, Lemma 1]. A σ-derivation of R is called X-inner if its extension to Q is equal to ad b,σ for some b ∈ Q. An automorphism of R is called X-inner if its extension to Q is equal to I u for some unit u ∈ Q. Following [14], define for each integer j the C-space Φ(j) def.= {u ∈ Q | ur = σj (r)u for all r ∈ Q}. j By [22, Chapter 3, Lemma 12.1], any 0 = u ∈ Φ(j) is a unit such that σ =Iu.We recall a very useful property due to Kharchenko and Popov [15]. Lemma 1.1. If δ is an X-outer σ-derivation, then σ(α)=α and either δ(α)=0 or σ = I δ(α) for each α ∈ C. Received by the editors June 29, 2010 and, in revised form, April 28, 2011. 2010 Mathematics Subject Classification. Primary 16S36, 16N60, 16W25, 16R50. Key words and phrases. Prime ring, (semi-)invariant polynomial, q-skew σ-derivation. The first two authors are members of the Mathematics Division, NCTS (Taipei Office). c 2012 American Mathematical Society Reverts to public domain 28 years from publication 3739 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3740 CHEN-LIAN CHUANG, TSIU-KWEN LEE, AND CHENG-KAI LIU Proof. Given α ∈ C and x ∈ R, δ(αx)=δ(α)x + σ(α)δ(x)andδ(xα)=δ(x)α + σ(x)δ(α) yield (σ(α) − α)δ(x)=σ(x)δ(α) − δ(α)x.Soσ(α)=α and σ(x)δ(α) − δ(α)x = 0 by the outerness of δ.Sincex ∈ R is arbitrary, δ(α) = 0 implies σ =Iδ(α) by [22, Chapter 3, Lemma 12.1]. We shall need the following generalization of Lemma 1.1. Lemma 1.2. Let δ be an X-outer σ-derivation. For a unit u ∈ Q,ifδ I u = γ I uδ, where γ ∈ C,thenσ(u)=γu and either δ(u)=0or σ = I u−1 δ(u). Proof. Expand 0 = δ(1) = δ(uu−1)=δ(u)u−1 + σ(u)δ(u−1). We see that δ(u−1)= −σ(u)−1δ(u)u−1. With this, we compute for x ∈ R −1 −1 −1 −1 δI u(x)=δ(uxu )=δ(u)xu + σ(u)δ(x)u + σ(u)σ(x)δ(u ) = δ(u)xu−1 + σ(u)δ(x)u−1 − σ(u)σ(x)σ(u)−1δ(u)u−1. −1 The assumption δI u(x)=γI uδ(x)=γuδ(x)u gives the identity δ(u)xu−1 + σ(u) − γu δ(x)u−1 − σ(u)σ(x)σ(u)−1δ(u)u−1 =0. But δ is X-outer. By [15, Theorem 1], σ(u)=γu and for x ∈ R, 0=δ(u)xu−1 − σ(u)σ(x)σ(u)−1δ(u)u−1 = δ(u)xu−1 − (γu)σ(x)(γu)−1δ(u)u−1 = δ(u)xu−1 − uσ(x)u−1δ(u)u−1 = u(u−1δ(u)x − σ(x)u−1δ(u))u−1. −1 So u δ(u) ∈ I σ.Ifδ(u) =0,then σ =Iu−1δ(u), as asserted. The Ore extension of R by a σ-derivation δ, denoted by R[t; σ, δ], is the set of n polynomial expressions a0 + a1t + ···+ ant ,wherea0,...,an ∈ R, with compo- nentwise addition and multiplication subjected to the rule tr = σ(r)t + δ(r)for all r ∈ R.WeformQ[t; σ, δ] analogously. Ore extensions have been extensively investigated in various directions. In the study of R[t; σ, δ], two crucially important notions are the notion of a right invariant polynomial and that of a semi-invariant polynomial. We omit “right” here for brevity and recall the definition below. Definition 1 ([16, 17, 18, 20]). We call f(t) ∈ Q[t; σ, δ] a cv-polynomial if it is associated to an automorphism τ of Q such that f(t)r − τ(r)f(t) ∈ Q for any r ∈ R. A cv-polynomial f(t)iscalledsemi-invariant if f(t)r − τ(r)f(t)=0 for all r ∈ R. A semi-invariant polynomial f(t) is called invariant if addition- ally f(t)t =(at + b)f(t)forsomea, b ∈ Q. A (semi-)invariant polynomial is called minimal if it is nonconstant and its degree is minimal among all nonconstant (semi-)invariant polynomials. The importance of these polynomials is based on the following: cv-polynomials determine R-stable homomorphisms from other Ore extensions of R into R[t; σ, δ] ([18] or [8, Theorem 12]). Semi-invariant polynomials determine the algebraic de- pendency of δ ([5]). Invariant polynomials determine the ideal structure of R[t; σ, δ] ([17, 20]). In particular, R[t; σ, δ] has nontrivial R-disjoint ideals iff R[t; σ, δ]has nonconstant (semi-)invariant polynomials ([20, Theorem 2.6]). For a simple ring R, R[t; σ, δ] is simple iff R[t; σ, δ] has no nonconstant (semi-)invariant polynomials. (Semi-)invariant polynomials for X-inner skew derivations can be found easily as follows. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INVARIANT POLYNOMIALS OF ORE EXTENSIONS 3741 def. Lemma 1.3. If δ = ad b,σ,whereb ∈ Q,thenπ(t) = t−b is the minimal invariant polynomial. Proof. Clearly, π(t)r = σ(r)π(t)forr ∈ R.Also, (t − b)t =(t − b)(t − b + b)=(t − b)2 +(t − b)b =(t − b)2 + σ(b)(t − b)=(t − b + σ(b))(t − b). So t − b is invariant. Let π(t) be a minimal monic semi-invariant polynomial of R[t; σ, δ]. It is shown in [20] that a factor of π(t)deg π(t) forms a minimal monic invariant polynomial. But for an ordinary derivation δ (namely for δ with σ =1),π(t) is also invariant when char R = 0 and either π(t) − c for some c ∈ Q or π(t)p is a minimal invariant polynomial when char R = p ≥ 2. Our aim here is to show that this is also true if δ is q-skew in a sense we now make clear. For any subset S of Q, define S(σ) def.= {r ∈ S | σ(r)=r},S(δ) def.= {r ∈ S | δ(r)=0} and S(σ,δ) def.= S(σ) ∩ S(δ). Definition. A σ-derivation δ is called q-skew, where 0 = q ∈ C(σ,δ),ifσ−1δσ = qδ. 2. Results Throughout the sequel, let δ be a q-skew σ-derivation and let π(t) be a minimal monic semi-invariant polynomial. In order for π(t) to exist, we assume that δ is quasi-algebraic [20, p. 147]. We aim to describe the minimal monic invariant polynomial in terms of π(t). By Lemma 1.3, we may consider X-outer δ.For X-outer δ, π(t) is given as follows. Theorem 2.1 ([4, Theorem 1] and [9, Theorem 15]). Let δ be an X-outer quasi- algebraic q-skew σ-derivation of R. Then there exists the least integer ν ≥ 1 such that qν =1and any minimal monic semi-invariant polynomial π(t) ∈ Q[t; σ, δ] is given by ⎧ ⎪ tν + b if char R =0, ⎨⎪ s−1 π(t)= s j ⎪ νp νp ≥ ⎩⎪ t + bj t + b if char R = p 2, j=0 s j where b ∈ Q and bj ∈ Φ(ν(p − p )) for 0 ≤ j ≤ s − 1.
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