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(σ,τ)-*-Jordan ideals in *-prime rings

(σ,τ)-*-Jordan ideals in *-prime rings AbstractLet R be a prime ring with involution ⋆{\star}, and let σ, τ be endomorphisms on R. For any x,y∈R{x,y\in R}, let (x,y)σ,τ=x⁢σ⁢(y)+τ⁢(y)⁢x{(x,y)_{\sigma,\tau}=x\sigma(y)+\tau(y)x}and Cσ,τ⁢(R)={x∈R∣x⁢σ⁢(y)=τ⁢(y)⁢x}{C_{\sigma,\tau}(R)=\{x\in R\mid x\sigma(y)=\tau(y)x\}}.An additive subgroup U of R is said to be a (σ,τ){(\sigma,\tau)}-right Jordan ideal (resp. (σ,τ){(\sigma,\tau)}-leftJordan ideal) of R if (U,R)σ,τ⊆U{(U,R)_{\sigma,\tau}\subseteq U}(resp. (R,U)σ,τ⊆U{(R,U)_{\sigma,\tau}\subseteq U}), and U is called a(σ,τ){(\sigma,\tau)}-Jordan ideal if U is both a (σ,τ){(\sigma,\tau)}-rightJordan ideal and a (σ,τ){(\sigma,\tau)}-left Jordan ideal of R. A (σ,τ){(\sigma,\tau)}-Jordan ideal Uof R is said to be a (σ,τ){(\sigma,\tau)}-⋆{\star}-Jordan ideal if U⋆=U{U^{\star}=U}.In the present paper, it is shown that if U is commutative, then R is commutative. The commutativity of R is also obtained if (U,U)σ,τ⊆Cσ,τ⁢(R){(U,U)_{\sigma,\tau}\subseteq C_{\sigma,\tau}(R)}. Some more results are obtained on the ⋆{\star}-prime ring with a characteristic different from 2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

(σ,τ)-*-Jordan ideals in *-prime rings

Ashraf, Mohammad; Parveen, Nazia
Georgian Mathematical Journal , Volume 26 (3): 9 – Sep 1, 2019
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References (15)

Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1572-9176
eISSN
1572-9176
DOI
10.1515/gmj-2017-0055
Publisher site
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Abstract

AbstractLet R be a prime ring with involution ⋆{\star}, and let σ, τ be endomorphisms on R. For any x,y∈R{x,y\in R}, let (x,y)σ,τ=x⁢σ⁢(y)+τ⁢(y)⁢x{(x,y)_{\sigma,\tau}=x\sigma(y)+\tau(y)x}and Cσ,τ⁢(R)={x∈R∣x⁢σ⁢(y)=τ⁢(y)⁢x}{C_{\sigma,\tau}(R)=\{x\in R\mid x\sigma(y)=\tau(y)x\}}.An additive subgroup U of R is said to be a (σ,τ){(\sigma,\tau)}-right Jordan ideal (resp. (σ,τ){(\sigma,\tau)}-leftJordan ideal) of R if (U,R)σ,τ⊆U{(U,R)_{\sigma,\tau}\subseteq U}(resp. (R,U)σ,τ⊆U{(R,U)_{\sigma,\tau}\subseteq U}), and U is called a(σ,τ){(\sigma,\tau)}-Jordan ideal if U is both a (σ,τ){(\sigma,\tau)}-rightJordan ideal and a (σ,τ){(\sigma,\tau)}-left Jordan ideal of R. A (σ,τ){(\sigma,\tau)}-Jordan ideal Uof R is said to be a (σ,τ){(\sigma,\tau)}-⋆{\star}-Jordan ideal if U⋆=U{U^{\star}=U}.In the present paper, it is shown that if U is commutative, then R is commutative. The commutativity of R is also obtained if (U,U)σ,τ⊆Cσ,τ⁢(R){(U,U)_{\sigma,\tau}\subseteq C_{\sigma,\tau}(R)}. Some more results are obtained on the ⋆{\star}-prime ring with a characteristic different from 2.

Journal

Georgian Mathematical Journal – de Gruyter

Published: Sep 1, 2019

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