Regularity of generalized derivations in $BCI$-algebras
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In this paper we study the regularity of inside (or outside) $(\theta,\phi)$-derivations in $BCI$-algebras $X$ and prove that let $\d:X\to X$ be an inside $(\map,\mbp)$-derivation of $X.$ If there exists $a\in X$ such that $\d(x)*\map(a)=0$, then $\d$ is regular for all $x\in X$. It is also shown that if $X$ is a $BCK$-algebra, then every inside (or outside) $(\map,\mbp)$-derivation of $X$ is regular. Furthermore the concepts of $\theta$-ideal, $\phi$-ideal and invariant inside (or outside) $(\theta,\phi)$-derivations of $X$ are introduced and their related properties are investigated. Finally we obtain the following result: If $\d:X\to X$ is an outside $(\map,\mbp)$-derivation of $X$, then $\d$ is regular if and only if every $\map$-ideal of $X$ is $\d$-invariant.
Abstract
In this paper we study the regularity of inside (or outside) $(\theta,\phi)$-derivations in $BCI$-algebras $X$ and prove that let $\d:X\to X$ be an inside $(\map,\mbp)$-derivation of $X.$ If there exists $a\in X$ such that $\d(x)*\map(a)=0$, then $\d$ is regular for all $x\in X$. It is also shown that if $X$ is a $BCK$-algebra, then every inside (or outside) $(\map,\mbp)$-derivation of $X$ is regular. Furthermore the concepts of $\theta$-ideal, $\phi$-ideal and invariant inside (or outside) $(\theta,\phi)$-derivations of $X$ are introduced and their related properties are investigated. Finally we obtain the following result: If $\d:X\to X$ is an outside $(\map,\mbp)$-derivation of $X$, then $\d$ is regular if and only if every $\map$-ideal of $X$ is $\d$-invariant.
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