Jordan triple (α,β)-higher ∗-derivations on semiprime rings
In this article, we define the following: Let N0{{\mathbb{N}}}_{0} be the set of all nonnegative integers and D=(di)i∈N0D={\left({d}_{i})}_{i\in {{\mathbb{N}}}_{0}} a family of additive mappings of a ∗\ast -ring RR such that d0=idR{d}_{0}=i{d}_{R}. DD is called a Jordan (α,β)\left(\a...
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-03-01
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| Series: | Demonstratio Mathematica |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/dema-2022-0213 |
| _version_ | 1797848714111352832 |
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| author | Ezzat O. H. |
| author_facet | Ezzat O. H. |
| author_sort | Ezzat O. H. |
| collection | DOAJ |
| description | In this article, we define the following: Let N0{{\mathbb{N}}}_{0} be the set of all nonnegative integers and D=(di)i∈N0D={\left({d}_{i})}_{i\in {{\mathbb{N}}}_{0}} a family of additive mappings of a ∗\ast -ring RR such that d0=idR{d}_{0}=i{d}_{R}. DD is called a Jordan
(α,β)\left(\alpha ,\beta )-higher
∗\ast -derivation (resp. a Jordan triple
(α,β)\left(\alpha ,\beta )-higher
∗\ast -derivation) of RR if dn(a2)=∑i+j=ndi(βj(a))dj(αi(a∗i)){d}_{n}\left({a}^{2})={\sum }_{i+j=n}{d}_{i}\left({\beta }^{j}\left(a)){d}_{j}\left({\alpha }^{i}\left({a}^{{\ast }^{i}})) (resp. dn(aba)=∑i+j+k=ndi(βj+k(a))dj(βk(αi(b∗i)))dk(αi+j(a∗i+j)){d}_{n}\left(aba)={\sum }_{i+j+k=n}{d}_{i}\left({\beta }^{j+k}\left(a)){d}_{j}\left({\beta }^{k}\left({\alpha }^{i}\left({b}^{{\ast }^{i}}))){d}_{k}\left({\alpha }^{i+j}\left({a}^{{\ast }^{i+j}}))) for all a,b∈Ra,b\in R and each n∈N0n\in {{\mathbb{N}}}_{0}. We show that the two notions of Jordan (α,β)\left(\alpha ,\beta )-higher ∗\ast -derivation and Jordan triple (α,β)\left(\alpha ,\beta )-higher ∗\ast -derivation on a 6-torsion free semiprime ∗\ast -ring are equivalent. |
| first_indexed | 2024-04-09T18:32:05Z |
| format | Article |
| id | doaj.art-8c2d650cdb8f4563b0c8743718e9f21b |
| institution | Directory Open Access Journal |
| issn | 2391-4661 |
| language | English |
| last_indexed | 2024-04-09T18:32:05Z |
| publishDate | 2023-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Demonstratio Mathematica |
| spelling | doaj.art-8c2d650cdb8f4563b0c8743718e9f21b2023-04-11T17:07:15ZengDe GruyterDemonstratio Mathematica2391-46612023-03-015611104111010.1515/dema-2022-0213Jordan triple (α,β)-higher ∗-derivations on semiprime ringsEzzat O. H.0Department of Mathematics, College of Science and Arts at Balgarn, University of Bisha, Sabt Al-Alaya(61985), Saudi ArabiaIn this article, we define the following: Let N0{{\mathbb{N}}}_{0} be the set of all nonnegative integers and D=(di)i∈N0D={\left({d}_{i})}_{i\in {{\mathbb{N}}}_{0}} a family of additive mappings of a ∗\ast -ring RR such that d0=idR{d}_{0}=i{d}_{R}. DD is called a Jordan (α,β)\left(\alpha ,\beta )-higher ∗\ast -derivation (resp. a Jordan triple (α,β)\left(\alpha ,\beta )-higher ∗\ast -derivation) of RR if dn(a2)=∑i+j=ndi(βj(a))dj(αi(a∗i)){d}_{n}\left({a}^{2})={\sum }_{i+j=n}{d}_{i}\left({\beta }^{j}\left(a)){d}_{j}\left({\alpha }^{i}\left({a}^{{\ast }^{i}})) (resp. dn(aba)=∑i+j+k=ndi(βj+k(a))dj(βk(αi(b∗i)))dk(αi+j(a∗i+j)){d}_{n}\left(aba)={\sum }_{i+j+k=n}{d}_{i}\left({\beta }^{j+k}\left(a)){d}_{j}\left({\beta }^{k}\left({\alpha }^{i}\left({b}^{{\ast }^{i}}))){d}_{k}\left({\alpha }^{i+j}\left({a}^{{\ast }^{i+j}}))) for all a,b∈Ra,b\in R and each n∈N0n\in {{\mathbb{N}}}_{0}. We show that the two notions of Jordan (α,β)\left(\alpha ,\beta )-higher ∗\ast -derivation and Jordan triple (α,β)\left(\alpha ,\beta )-higher ∗\ast -derivation on a 6-torsion free semiprime ∗\ast -ring are equivalent.https://doi.org/10.1515/dema-2022-0213semiprime ringsinvolutionsderivationsjordan ∗-derivationshigher derivationsprimary 16w25secondary 16w1039b0516n6016u80 |
| spellingShingle | Ezzat O. H. Jordan triple (α,β)-higher ∗-derivations on semiprime rings Demonstratio Mathematica semiprime rings involutions derivations jordan ∗-derivations higher derivations primary 16w25 secondary 16w10 39b05 16n60 16u80 |
| title | Jordan triple (α,β)-higher ∗-derivations on semiprime rings |
| title_full | Jordan triple (α,β)-higher ∗-derivations on semiprime rings |
| title_fullStr | Jordan triple (α,β)-higher ∗-derivations on semiprime rings |
| title_full_unstemmed | Jordan triple (α,β)-higher ∗-derivations on semiprime rings |
| title_short | Jordan triple (α,β)-higher ∗-derivations on semiprime rings |
| title_sort | jordan triple α β higher ∗ derivations on semiprime rings |
| topic | semiprime rings involutions derivations jordan ∗-derivations higher derivations primary 16w25 secondary 16w10 39b05 16n60 16u80 |
| url | https://doi.org/10.1515/dema-2022-0213 |
| work_keys_str_mv | AT ezzatoh jordantripleabhigherderivationsonsemiprimerings |