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On near-ring ideals with \((\sigma,\tau)\)-derivation. (English) Zbl 1156.16030
Summary: Let \(N\) be a \(3\)-prime left near-ring with multiplicative center \(Z\); a \((\sigma,\tau)\)-derivation \(D\) on \(N\) is defined to be an additive endomorphism satisfying the product rule \(D(xy)=\tau(x)D(y)+D(x)\sigma(y)\) for all \(x,y\in N\), where \(\sigma\) and \(\tau\) are automorphisms of \(N\). A nonempty subset \(U\) of \(N\) will be called a semigroup right ideal (resp. semigroup left ideal) if \(UN\subset U\) (resp. \(NU\subset U\)) and if \(U\) is both a semigroup right ideal and a semigroup left ideal, it be called a semigroup ideal.
We prove the following results: Let \(D\) be a \((\sigma,\tau)\)-derivation on \(N\) such that \(\sigma D=D\sigma\), \(\tau D=D\tau\). (i) If \(U\) is a semigroup right ideal of \(N\) and \(D(U)\subset Z\), then \(N\) is a commutative ring. (ii) If \(U\) is a semigroup ideal of \(N\) and \(D^2(U)=0\) then \(D=0\). (iii) If \(a\in N\) and \([D(U),a]_{\sigma,\tau}=0\) then \(D(a)=0\) or \(a\in Z\).
MSC:
16Y30 Near-rings
16W25 Derivations, actions of Lie algebras
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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