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Rings with annihilator conditions on multilinear polynomials. (English) Zbl 0855.16029
Let \(K\) be a commutative ring with \(1\), \(R\) a \(K\)-algebra with no nonzero nil right ideal, and \(f(X_1,\dots,X_m)\) a multilinear polynomial over \(K\). The authors prove that if a right ideal \(T\) of \(R\) and \(y\in R\) satisfy \(yf(a_1,\dots,a_m)^n=0\) for all \(a_i\in T\) and \(n=n(a_1,\dots,a_m)\), then \(yf(a_1,\dots,a_m)T=0\). With the same hypothesis, but for a left ideal \(L\) of \(R\), the conclusion is that \(yLf(a_1,\dots,a_m)=0\). The authors observe that the arguments given allow one to assume that \(R\) is any semiprime \(K\)-algebra if the power \(n\) is a fixed integer rather than dependent on the choice of the \(a_i\in L\).

MSC:
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16N60 Prime and semiprime associative rings
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