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Left annihilators characterized by GPIs. (English) Zbl 0845.16016
Let $$R$$ be a semiprime ring with extended centroid $$C$$ and right Utumi (maximal) quotient ring $$U$$. The author gives a clever argument using the theory of orthogonal completions to prove the following interesting result: If $$A$$ and $$B$$ are right ideals of $$R$$ and $$S$$ is a subring satisfying $$R\subseteq S\subseteq U$$, then $$A$$ and $$B$$ satisfy the same generalized polynomial identities with coefficients in $$SC$$ if and only if $$l_S(A)=l_S(B)$$, where $$l_S(T)=\{x\in S\mid xt=0$$ for all $$t\in T\}$$. In particular, if $$Q$$ is the two sided Utumi quotient ring of $$R$$, or the symmetric Martindale quotient ring, it follows that $$A$$ and $$R$$ satisfy the same generalized polynomial identities over $$Q$$ when $$l_R(A)=0$$.

##### MSC:
 16R50 Other kinds of identities (generalized polynomial, rational, involution) 16N60 Prime and semiprime associative rings 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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##### References:
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