Partial Hopf actions on generalized matrix algebras

<p>Given a unital commutative ring $ \mathscr{R} $, $ (\mathscr{A}, \mathscr{B}) $ and $ (\mathscr{B}, \mathscr{A}) $ are bimodules of $ \mathscr{M} $ and $ \mathscr{N} $, respectively, where $ \mathscr{A}, \mathscr{B} $ are unitals $ \mathscr{R}- $algebras. The $ \mathscr{R}- $algebra $ \mathscr{G} = $ $ \mathscr{G}(\mathscr{A}, \mathscr{M}, \mathscr{N}, \mathscr{B}) $ is a generalized matrix algebra described by the Morita context $ (\mathscr{A}, \mathscr{B}, \mathscr{M}, \mathscr{N}, \zeta_{\mathscr{M}\mathscr{N}}, \chi_{\mathscr{N}\mathscr{M}}) $. The present study investigated the structure of Lie (Jordan) $ \sigma- $centralizers at the zero products on order two generalized matrix algebra and established that each Jordan $ \sigma- $centralizer at the zero products is a $ \sigma- $centralizer at the zero product on order two generalized matrix algebra. We also provided sufficient and necessary conditions under which a Lie $ \sigma- $centralizer at the zero product is proper on an order two generalized matrix algebra.</p>

In this article, we introduce the notion of Lie triple centralizer as follows. Let be an algebra, and be a linear mapping. We say that ϕ is a Lie triple centralizer whenever for all . Then we characterize the general form of Lie triple centralizers on a generalized matrix algebra and under some mild conditions on we present the necessary and sufficient conditions for a Lie triple centralizer to be proper. As an application of our results, we characterize generalized Lie triple derivations on generalized matrix algebras.

In this article, the additivity of higher multiplicative mappings, i.e., Jordan mappings, on generalized matrix algebras are studied. Also, the definition of Jordan higher triple product homomorphism is introduced and its additivity on generalized matrix algebras is studied.

Let be a commutative ring with unity, be -algebras, be ()-bimodule and be ()-bimodule. The -algebra is a generalized matrix algebra defined by the Morita context . In this article, we study multiplicative generalized Lie triple derivation on generalized matrix algebras and prove that every multiplicative generalized Lie triple derivation on a generalized matrix algebra has the standard form.

Let be a commutative ring with unity and be a generalized matrix algebra. In this article, we give the structure of Lie triple derivation on a generalized matrix algebra and prove that under certain appropriate assumptions on is proper, i.e., where δ is a derivation on and χ is a mapping from into its center which annihilates all second commutators in i.e., for all

Let [Formula: see text] be a generalized matrix algebra defined by the Morita context. We consider [Formula: see text] as a superalgebra following the method proposed by Ghahramani and Heidari Zadeh. This paper investigates the problem of describing the form of Jordan superderivations on generalized matrix algebras. It turns out that in generalized matrix algebras, there exist Jordan superderivations that are not superderivations. Additionally, we show that every generalized Jordan superderivation on a class of generalized matrix algebras can be expressed as the sum of a generalized superderivation and an anti-superderivation. Consequently, we characterize (generalized) Jordan superderivation on a triangular algebra. In particular, we deduce that any (generalized) Jordan superderivation of an upper triangular algebra, under certain conditions, is a (generalized) derivation.

Let be a generalized matrix algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and is the trace of . We describe the form of satisfying the condition for all . The question of when has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

Let G be a generalized matrix algebra over a commutative ring R, q: G × G ! G be an R-bilinear mapping and Tq: : G ! G be a trace of q. We describe the form of Tq satisfying the condition Tq(G)G = GTq(G) for all G 2 G. The question of when Tq has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie isomorphism of G to be almost standard. As applications we characterize Lie isomorphisms of full matrix algebras, of triangular algebras and of certain uni- tal algebras with nontrivial idempotents. Some further research topics related to current work are proposed at the end of this article.

<abstract><p>In this article, we proved that each nonlinear higher anti-derivable mapping on generalized matrix algebras is automatically additive. As for its applications, we find a similar conclusion on triangular algebras, full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras and factor von Neumann algebras respectively.</p></abstract>

Let be a commutative ring with unit element, be -algebras, be an -bimodule, and be a -bimodule. The -algebra is a generalized matrix algebra defined by the Morita context In this article, we study Lie (Jordan) centralizer on generalized matrix algebras and obtain the necessary and sufficient conditions for a Lie centralizer map to be proper. Further, we prove that every Jordan centralizer is a centralizer on generalized matrix algebras under certain assumptions.

be a generalized matrix algebra defined by the Morita context

Let 𝒭 be a commutative ring with unity, 𝒜, 𝒝 be 𝒭-algebras, 𝒨 be (𝒜, 𝒝)-bimodule and 𝒩 be (𝒝, 𝒜)-bimodule. The 𝒭-algebra 𝒢 = 𝒢(𝒜, 𝒨, 𝒩, 𝒝) is a generalized matrix algebra defined by the Morita context (𝒜, 𝒝, 𝒨, 𝒩, ξ 𝒨𝒩 , Ω 𝒩𝒨 ). In this article, we study Jordan σ- derivations on generalized matrix algebras.

Let be a commutative ring with identity, be -algebras, be an -bimodule and be a -bimodule. The -algebra is a generalized matrix algebra defined by the Morita context In this article, we give the structure of generalized Lie triple derivations GL on generalized matrix algebras and prove that under certain restrictions GL can be written as where Δ is a generalized derivation and χ is a central valued mapping.

Commuting mappings of generalized matrix algebras

In this article, we study n-derivations on generalized matrix algebras under certain restrictions and find that every n-derivation is an extremal n-derivation on generalized matrix algebras.