Derivations with invertible values in rings with involution

Giambruno, Antonio; Misso, P.; Milies, César Polcino

doi:10.2140/pjm.1986.123.47

Cited by 8 publications

(11 citation statements)

References 1 publication

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“…Since any element which is invertible in A (+) is also invertible in A, then d is a derivation with invertible values of associative algebra A, so, by [1] A is either a division algebra D, or a D 2 , the 2 × 2 matrix algebra over a division algebra D. [6] it follows that if dim Z(A) A > 4, then < H(A, * ) >= A. Wedderburn-Artin theorem implies that if dim Z(A) A ≤ 4, then A is either a division algebra over Z(A) or Z(A) 2 , which correspondingly matches the cases 1) and 2), so from now on we may assume that d can be extended to a derivation of A. As a derivation of A, d also has invertible values, so by [8] A is an algebra of type 1), 2) or 3). The lemma is now proved.…”

Section: Preliminariesmentioning

confidence: 99%

Jordan algebras admitting derivations with invertible values

Kaygorodov

Lopatin

Popov

2017

Communications in Algebra

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The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of associative rings that admit derivations with invertible values. Later, the results of this paper were generalized in many cases, for example, for generalized derivations, associative superalgebras, alternative algebras and many others. The present work is dedicated to description of all Jordan algebras admitting derivations with invertible values.

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Section: Preliminariesmentioning

confidence: 99%

Jordan algebras admitting derivations with invertible values

Kaygorodov

Lopatin

Popov

2017

Communications in Algebra

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“…They also characterized those division rings such that a 2 × 2 matrix ring over them has an inner derivation with invertible values. Further, associative rings with derivations with invertible values (and also their generalizations) were discussed in variety of works (see, for instance, [2]- [6]). So, in [2], semiprime associative rings with involution, allowing a derivation with invertible values on the set of symmetric elements, were given an examination.…”

Section: Alternative Algebras With Derivations With Invertible Valuesmentioning

confidence: 99%

“…Further, associative rings with derivations with invertible values (and also their generalizations) were discussed in variety of works (see, for instance, [2]- [6]). So, in [2], semiprime associative rings with involution, allowing a derivation with invertible values on the set of symmetric elements, were given an examination. In work [3] Bergen and Carini determined the associative rings admitting a derivation with invertible values on some non-central Lie ideal.…”

Section: Alternative Algebras With Derivations With Invertible Valuesmentioning

confidence: 99%

Alternative algebras admitting derivations with invertible values and invertible derivations

Kaygorodov¹,

Popov²

2014

Izv. Math.

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“…Берген, И. Н. Херстейн и Ч. Лански описали структуру ассоциативных колец, допускающих дифференцирования с обратимыми значениями. В дальнейшем результаты этой работы в ассоциативном случае получили обобщение в [2]- [6].…”

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“…В дальнейшем ассоциативные кольца, допускающие дифференцирования с обратимыми значениями (и их обобщения), были рассмотрены в [2]- [6]. Так, в [2] было описано строение полупростых ассоциативных колец с инволюцией, допускающих дифференцирование, значения которого на симметрических элементах кольца являются обратимыми. В работе [3] описаны ассоциативные кольца с дифференцированием, значения которого на некотором нецентральном лиевом идеале обратимы.…”

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