A one-sided version of Posner's second theorem on multilinear polynomials
Bull. Korean Math. Soc. 2005 Vol. 42, No. 4, 679-690
Published online December 1, 2005
Vincenzo De Filippis
Universita di Messina
Abstract : Let $K$ be a commutative ring with unity, $R$ a prime $K$-algebra of characteristic different from 2, $d$ a non-zero derivation of $R$, $I$ a non-zero right ideal of $R$, $f(x_1,\ldots,x_n)$ a multilinear polynomial in n non-commuting variables over $K$, $a\in R$. Supppose that, for any $x_1,\ldots,x_n\in I$, $a[d(f(x_1,\ldots,x_n)),f(x_1,\ldots,x_n)]=0$. If $[f(x_1,\ldots,x_n),x_{n+1}]x_{n+2}$ is not an identity for $I$ and $$S_4(I,I,I,I)I\neq 0,$$ then $aI=ad(I)=0.$
Keywords : prime rings, derivations, generalized polynomial identities
MSC numbers : Primary 16N60; Secondary 16W25
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