A note on n-Jordan homomorphisms
Uložené v:
| Názov: | A note on n-Jordan homomorphisms |
|---|---|
| Autori: | M. El Azhari |
| Zdroj: | Математичні Студії, Vol 62, Iss 1, Pp 77-80 (2024) |
| Informácie o vydavateľovi: | Ivan Franko National University of Lviv, 2024. |
| Rok vydania: | 2024 |
| Predmety: | homomorphism, n-jordan homomorphism, QA1-939, n-homomorphism, jordan homomorphism, Mathematics |
| Popis: | {Let $ A, B $ be two rings and $ n\geqslant 2 $ be an integer. An additive map $ h\colon A\rightarrow B $ is called an $n$-Jordan homomorphism if $ h(x^{n})=h(x)^{n} $ for all $ x\in A;$ $h$ is called an n-homomorphism or an anti-$n$-homomorphism if $ h(\prod_{i=1}^{n}x_{i})=\prod_{i=1}^{n} h(x_{i})$ or $ h(\prod_{i=1}^{n}x_{i})=\prod_{i=0}^{n-1} h(x_{n-i}), $ respectively, for all $ x_{1},...,x_{n}\in A. $} {We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let $n\geq 2$ be an integer and $h$ be an $n-$Jordan homomorphism from a ring $A$ into a ring $B$ of characteristic greater than $n$. Suppose further that $A$ has a unit $e$, then $h = h(e)\tau$, where $h(e)$ is in the centralizer of $h(A)$ and $\tau$ is a Jordan homomorphism.} {By using this variation, we deduce the following result of G. An: Let $A$ and $B$be two rings, where $A$ has a unit and $B$ is of characteristic greater than an integer $n \geq 2$. If every Jordan homomorphism from $A$ into $B$ is a homomorphism (anti-homomorphism), then every $n-$Jordan homomorphism from $A$ into $B$ is an $n$-homomorphism (anti-$n$-homomorphism).As a consequence of an appropriate lemma, we also obtain the following resultof E. Gselmann: Let $A, B$ be two commutative rings and $B$ is of characteristic greater than an integer $n\geq 2$. Then every $n$-Jordan homomorphism from $A$ into$B$ is an $n-$homomorphism.} |
| Druh dokumentu: | Article |
| ISSN: | 2411-0620 1027-4634 |
| DOI: | 10.30970/ms.62.1.77-80 |
| Prístupová URL adresa: | https://doaj.org/article/31fb85f9b3fa471aa3e879e46127b481 |
| Rights: | CC BY NC ND |
| Prístupové číslo: | edsair.doi.dedup.....20be945d4e381a56697d3c96705aeecd |
| Databáza: | OpenAIRE |
Full Text Finder 
Nájsť tento článok vo Web of Science | Abstrakt: | {Let $ A, B $ be two rings and $ n\geqslant 2 $ be an integer. An additive map $ h\colon A\rightarrow B $ is called an $n$-Jordan homomorphism if $ h(x^{n})=h(x)^{n} $ for all $ x\in A;$ $h$ is called an n-homomorphism or an anti-$n$-homomorphism if $ h(\prod_{i=1}^{n}x_{i})=\prod_{i=1}^{n} h(x_{i})$ or $ h(\prod_{i=1}^{n}x_{i})=\prod_{i=0}^{n-1} h(x_{n-i}), $ respectively, for all $ x_{1},...,x_{n}\in A. $} {We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let $n\geq 2$ be an integer and $h$ be an $n-$Jordan homomorphism from a ring $A$ into a ring $B$ of characteristic greater than $n$. Suppose further that $A$ has a unit $e$, then $h = h(e)\tau$, where $h(e)$ is in the centralizer of $h(A)$ and $\tau$ is a Jordan homomorphism.} {By using this variation, we deduce the following result of G. An: Let $A$ and $B$be two rings, where $A$ has a unit and $B$ is of characteristic greater than an integer $n \geq 2$. If every Jordan homomorphism from $A$ into $B$ is a homomorphism (anti-homomorphism), then every $n-$Jordan homomorphism from $A$ into $B$ is an $n$-homomorphism (anti-$n$-homomorphism).As a consequence of an appropriate lemma, we also obtain the following resultof E. Gselmann: Let $A, B$ be two commutative rings and $B$ is of characteristic greater than an integer $n\geq 2$. Then every $n$-Jordan homomorphism from $A$ into$B$ is an $n-$homomorphism.} |
|---|---|
| ISSN: | 24110620 10274634 |
| DOI: | 10.30970/ms.62.1.77-80 |