Lie structure of prime rings of characteristic 2

Lanski, Charles; Montgomery, Susan

doi:10.2140/pjm.1972.42.117

Cited by 142 publications

(33 citation statements)

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“…There is a particular interest in Lie ideals U such that u 2 ∈ U for every u ∈ U . Such a distinction already appears in several papers involving usual derivations and Lie ideals ( [1], [2], [8]). …”

Section: Introductionmentioning

confidence: 92%

Higher Derivations on Lie Ideals

Haetinger¹

2002

TEMA - Tendências Em Matemática Aplicada E Computacional

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Abstract. In this paper we present a brief proof of a recently proved result [5, Corollary 1.4]. The main result states that if R is a prime ring of characteristic different of 2 and U is a Lie ideal of R where U ⊂ Z(R), the center of R, u 2 ∈ U for all u ∈ U , and D is a Jordan higher derivation of U into R, then D is a higher derivation of U into R. This result extends a theorem of Awtar [1].

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Section: Introductionmentioning

confidence: 92%

Higher Derivations on Lie Ideals

Haetinger¹

2002

TEMA - Tendências Em Matemática Aplicada E Computacional

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“…Then [U, U ] = 0 in R = R/P . Since charR = 2, we must have that U is central by [10,Theorem 4]. That is, [U, R] ⊆ P .…”

Section: Suppose That D(p )mentioning

confidence: 99%

A result on derivations

Lee

Lin

1996

Proc. Amer. Math. Soc.

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Abstract. Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a ∈ R. Suppose that ad(u) n = 0 for all u ∈ U, where n is a fixed positive integer. Then ad(I) = 0 for I the ideal of R generated by [U, U ] and if R is 2-torsion free, then ad(U ) = 0. Furthermore, R is a subdirect sum of semiprime homomorphic images R 1 and R 2 with derivations d 1 and d 2 , induced canonically by d, respectively such that ad 1 (R 1 ) = 0 and the image of U in R 2 is commutative (central if R is 2-torsion free), where a denotes the image of a in R 1 . Moreover, if U = R, then ad(R) = 0. This gives Bresar's theorem without the (n − 1)!-torsion free assumption on R. In [8] I. N. Herstein proved that if R is a prime ring and d is an inner derivation of R such that d(x)n = 0 for all x ∈ R and n a fixed integer, then d = 0. In [6] A. Giambruno and I. N. Herstein extended this result to arbitrary derivations in semiprime rings. In [2] L. Carini and A. Giambruno proved that if R is a prime ring with a derivation d such that d (x) n(x) = 0 for all x ∈ U, a Lie ideal of R, then d(U ) = 0 when R has no nonzero nil right ideals, char R = 2 and the same conclusion holds when n(x) = n is fixed and R is a 2-torsion free semiprime ring. Using the ideas in [2] and the methods in [5] C. Lanski [11] removed both the bound on the indices of nilpotence and the characteristic assumptions on R.In [1] M. Bresar gave a generalization of the result due to I. N. Herstein and A. Giambruno [6] in another direction. Explicitly, he proved the theorem: Let R be a semiprime ring with a derivation d, a ∈ R. If ad(x) n = 0 for all x ∈ R, where n is a fixed integer, then ad(R) = 0 when R is an (n − 1)!-torsion free ring. The present paper is then motivated by Bresar's result and by Lanski's paper [11]. We prove Bresar's result without the assumption of (n − 1)!-torsion free on R. In fact, we study the Lie ideal case as given in [11] and then obtain Bresar's result as the corollary to our main result. More precisely, we shall prove the following Main Theorem. Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a ∈ R. Suppose that ad(u) n = 0 for all u ∈ U , where n is a fixed integer. Then ad(I) = 0 for I the ideal of R generated by [U, U ] and if R is 2-torsion free, then ad(U ) = 0.Furthermore, R is a subdirect sum of semiprime homomorphic images R 1 and R 2 with derivations d 1 and d 2 , induced canonically by d, respectively such that

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“…120]. The possibility Q {i) + Q* c Z(R/Q*) and repeated use of [11;Lemma 7,p. 120] force Q + Q* c Z(R/Q*), which in turn means that R/Q* is commutative.…”

Section: Is a Lie Ideal In R/q* It Follows That Either A + Q*az(r/q*mentioning

confidence: 99%