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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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