On the trace of permuting tri-derivations on rings

Q3 Mathematics Matematychni Studii Pub Date : 2022-10-31 DOI:10.30970/ms.58.1.20-25
D. Yılmaz, H. Yazarli
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引用次数: 0

Abstract

In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.Let $R$ be a $2,3$-torsion free prime ring and $F:R\times R\times R\rightarrow R$ be a permuting tri-derivation with trace $f$, $ d:R\rightarrow R$ be a derivation. In particular, the following assertions have been proved:1) if $[d(r),r]=f(r)$ for all $r\in R$, then $R$ is commutative or $d=0$ (Theorem 1);\ 2) if $g:R\rightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $r\in R$, then $F=0$ or $d=0$ (Theorem 2); 3) if $F(d(r),r,r)=f(r)$ for all $r\in R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)\circ f(r)=0$ for all $r\in R$ (Theorem 3). In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:R\times R\times R\rightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $r\in R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)\circ f_{2}(r)=0$ for all $r\in R$ (Theorem 4).
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环上置换三导的迹
本文研究了素数环和半素数环上的导数、置换三导数和自同态的相互影响。设$R$是一个2,3$无扭素环,$F:R\乘以R\右列R$是一个有迹$F $的置换三重导数,$ d:R\右列R$是一个导数。特别地,证明了下列断言:1)如果$[d(r),r]=f(r)$对于r $中的所有$r\是交换的或$d=0$(定理1);2)如果$g: r\右列r $是自同态使得$ f(d(r),r,r)=g(r)$对于r $中的所有$r\是自同态,则$ f =0$或$d=0$(定理2);3)如果$ F (d (r), r, r) = F (r) $ r \ r美元,然后(i) $ F = 0美元或美元d = 0美元,美元(ii) $ $ d (r) \保监会F (r) = 0中所有$ r \ r美元(定理3)。在另一方面,如果存在交换tri-derivations $ F {1}, F{2}: \乘以r \ r \ rightarrow r F {1} $, $ (F {2} (r), r, r) = F {1} (r)为所有r \ r美元,美元$ F{1} $和$ % F {2} $ $ F{1} $的痕迹和F{2},美元,那么美元(i) $ $ F {1} = 0 F{2} = 0美元或美元,美元(ii) $ $ F {1} (r) F{2} \保监会(r) = 0中所有$ r \ r美元(定理4)。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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