Nonlinear derivations of incidence algebras

Yang, Yichuan

doi:10.1007/s10474-019-01016-2

Cited by 3 publications

(2 citation statements)

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“…The most fundamental result in this direction is due to W. S. Martindale III [35] who proved that every bijective multiplicative mapping from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive. Later, a number of authors considered the Jordan-type product or Lie-type product and proved that, on certain associative algebras or rings, bijective mappings which preserve any of those products are automatically additive, see [17,45,48,[50][51][52].…”

Section: Related Topics For Further Researchmentioning

confidence: 99%

“…Since then, automorphisms, involutions, Lie derivation and Jordan derivations of finitary incidence algebras and related topics have been increasingly significant, see [2, 6-9, 11, 13-16, 23-27, 29, 30, 38-40, 43, 47, 49, 55] and the references therein. Recently, the nonlinear derivations and nonlinear Lie derivations of incidence algebras were studied in [50][51][52]. A new phenomenon shows that for general incidence algebras, there exist nonlinear Lie derivations which are not proper.…”

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Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

Yang,

Wei

2021

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This is a continuation of our earlier works [29,51,52] with respect to (non-)linear Lie-type derivations of finitary incidence algebras. Let X be a pre-ordered set, R be a 2-torsionfree and (n − 1)torsionfree commutative ring with identity, where n ≥ 2 is an integer. Let F I(X, R) be the finitary incidence algebra of X over R. In this paper, a complete clarification is obtained for the structure of nonlinear Lie-type derivations of F I(X, R). We introduce a new class of derivations on F I(X, R) named inner-like derivations, and prove that each nonlinear Lie n-derivation on F I(X, R) is the sum of an inner-like derivation, a transitive induced derivation and a quasi-additive induced Lie n-derivation. Furthermore, if X is finite, we show that a quasi-additive induced Lie n-derivation can be expressed as the sum of an additive induced Lie derivation and a central-valued map annihilating all (n − 1)-th commutators. We also provide a sufficient and necessary condition such that every nonlinear Lie n-derivation of F I(X, R) is of proper form. Some related topics for further research are proposed in the last section of this article.Let A be an associative algebra over a commutative ring R. Let [ , ] and • be the Lie product and Jordan product respectively, i.e., [x, y] = xy − yx and x • y = xy + yx for all x, y ∈ A. Then (A, [ , ]) is a Lie algebra and (A, •) is a Jordan algebra. It is a fascinating topic to study the connection between the associative, Lie and Jordan structures of A. In this field, there are two classes of important algebraic maps: algebra homomorphisms and differential operators. For example Jordan homomorphisms, Lie homomorphisms, Jordan derivations and Lie derivations. In the AMS Hour Talk of 1961, Herstein proposed many problems

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