Lanski, Charles A note on GPIS and their coefficients. (English) Zbl 0608.16022 Proc. Am. Math. Soc. 98, 17-19 (1986). 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 Cited in 1 ReviewCited in 5 Documents MSC: 16Rxx Rings with polynomial identity 16N60 Prime and semiprime associative rings 16P50 Localization and associative Noetherian rings Keywords:prime ring; generalized polynomial identity; multilinear GPI; Martindale quotient ring; free product; extended centroid PDFBibTeX XMLCite \textit{C. Lanski}, Proc. Am. Math. Soc. 98, 17--19 (1986; Zbl 0608.16022) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.