Automorphic-differential identities in rings

Bergen, Jeffrey

doi:10.1090/s0002-9939-1989-0967482-3

Cited by 9 publications

(6 citation statements)

References 8 publications

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“…Yanai deduced that if any such ϕ vanishes on a right ideal of R with the zero left annihilator then ϕ also vanishes on M (Theorem 5 [25]). This generalizes Bergen [2]. Note that [3] is also along this line.…”

Section: Resultssupporting

confidence: 53%

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Nonlinear identities with skew derivations

Chuang

2016

Journal of Algebra

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MSC: 16N60 16R50 16S40 16T15 16W20 16W22 16W25 16W25 Keywords: Prime ring Automorphism Skew derivation Identity Coalgebras Pointed coalgebras The complete ring of quotients Utumi quotient ringLet R be a prime ring with the extended centroid c. Suppose that R is acted by a pointed coalgebra with group-like elements acting as automorphisms of R. A generalized polynomial with variables acted by the coalgebra is called an identity if it vanishes on R. We prove the following:(1) If c is a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R with variables acted by Frobenius automorphisms.(2) If c is not a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R. With this, we extend Yanai's result [25] to "nonlinear identities". These are actually special instances of our Theorems 1 and 2 below respectively, which extend Kharchenko's theory of differential identities [14,15] to the context of expansion closed word sets introduced in [6].

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Section: Resultssupporting

confidence: 53%

“…and χ S Δ (x, y, z) = 0 by (2 ). Write ψ S (x, y) := ψ S Δ (x, y) and χ S (x, y, z) := χ S Δ (x, y, z) for short.…”

Section: Proof Of Lemmamentioning

confidence: 99%

Nonlinear identities with skew derivations

Chuang

2016

Journal of Algebra

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“…In [B,Question 1], Bergen also asked whether Theorem A holds even if R is semiprime, and in [O], A. Ouarit gave an affirmative answer to this problem. We extend this result to an endomorphism with skew-derivations, using a different proof from [O], in Section 4.…”

Section: Theorem a ([B Theorem 1]) Let A Be A Right Ideal Of A Primmentioning

confidence: 99%

Automorphic-differential identities and actions of pointed coalgebras on rings

Yanai

1998

Proc. Amer. Math. Soc.

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Abstract. In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let R be a ring, R F its left Martindale quotient ring and A a right ideal of R having no nonzero left annihilator. (1) Let C be a pointed coalgebra which measures R such that the group-like elements of C act as automorphisms of R. If R is prime and ξ · A = 0 for ξ ∈ R#C, then ξ · R = 0. Furthermore, if the action of C extends to R F and if ξ ∈ R F #C such that ξ · A = 0, then ξ · R F = 0. (2) Let f be an endomorphism of R F given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If R is semiprime and f (A) = 0, then f(R) = 0.

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“…The problem was also mentioned in [4,9,11]. The best result of the conjecture is the following: A ring R is said to be of bounded index m if m is a positive integer such that x m = 0 for all nilpotent elements x ∈ R. Beidar and Mikhalëv proved the theorem: Let R be a ring of bounded index m such that the additive order of every nonzero torsion element of R, if any, is strictly larger than m. Then all minimal prime ideals of R are invariant under all derivations of R. (See [2] or [3,Theorem 8.16]. )…”

Section: Introductionmentioning

confidence: 99%