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A note on GPIS and their coefficients. (English) Zbl 0608.16022

A prime ring R satisfying a GPI (generalized polynomial identity), satisfies a multilinear GPI having all its coefficients in R. All R bimodules in the Martindale quotient ring of R satisfy the same multilinear GPI’s. These two facts are shown in the note.
The proof is given in a more general setting. Let Q be the Martindale quotient ring of R and F the free product over C, the extended centroid of R, of Q and the free algebra over C on the set \(X=\{x_ 1,x_ 2,...\}\). An R-subbimodule T (\(\neq 0)\) of Q satisfies a Q-GPI if for some \(f(x_ 1,x_ 2,...)\) in F-\(\{\) \(0\}\), \(f(t_ 1,t_ 2,...)=0\) for all substitutions \(t_ i\) in T for \(x_ i\). The above mentioned results follow from the following more general theorems: 1. Let T be a nonzero R- subbimodule of Q. If \(f\in F\) is a multilinear and homogeneous Q-GPI for T, then f is a Q-GPI for Q. 2. Let T and S be any nonzero R-subbimodules of Q. If T satisfies a Q-GPI, then T satisfies some Q-GPI having all its coefficients in S.
Reviewer: Jan Van Geel

MSC:

16Rxx Rings with polynomial identity
16N60 Prime and semiprime associative rings
16P50 Localization and associative Noetherian rings
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References:

[1] I. N. Herstein, Rings with involution, The University of Chicago Press, Chicago, Ill.-London, 1976. Chicago Lectures in Mathematics. · Zbl 0343.16011
[2] Wallace S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576 – 584. · Zbl 0175.03102 · doi:10.1016/0021-8693(69)90029-5
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