Damiano, Robert F.; Shapiro, Jay Twisted polynomial rings satisfying a polynomial identity. (English) Zbl 0558.16001 J. Algebra 92, 116-127 (1985). Let R be a noetherian ring with prime radical N. First, the authors investigate conditions for twisted polynomial rings over R to be FBN. They show that if R[x,\(\sigma\) ] is an FBN ring for \(\sigma\) an automorphism of R, then \({\bar \sigma}{}^ m_{Z(R/N)}=1\) for some m where \({\bar \sigma}{}_{Z(R/N)}\) is the induced automorphism of the center of R/N. When R is semiprime PI a stronger result is achieved which deals with certain endomorphisms \(\sigma\). In fact, if R is PI with automorphism \(\sigma\), they show that R[x,\(\sigma\) ] is noetherian PI precisely when \({\bar \sigma}{}^ m=1_{Z(R/N)}\) for some m. In between they show that if R is algebraic over its center and \(\sigma\) is an automorphism such that, for some m, the induced automorphism \({\bar \sigma}{}^ m\) of R/N is conjugation by a unit, then R[x,\(\sigma\) ] is FBN. The rest of the paper is concerned with localization of twisted polynomial rings. For example if R is FBN with enough clans, sufficient conditions are given for a finite normalizing extension of R to be FBN with enough clans. This has two interesting corollaries giving conditions on an automorphism \(\sigma\) for R[x,\(\sigma\) ] to be FBN (PI) with enough clans when R is integral over its center (affine). The paper ends with two examples where R[x,\(\sigma\) ] does not have enough clans - in the second example R is an artinian PI ring with only one proper nonzero ideal. Reviewer: R.Gordon Cited in 1 ReviewCited in 9 Documents MSC: 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16Rxx Rings with polynomial identity 16P40 Noetherian rings and modules (associative rings and algebras) 16Dxx Modules, bimodules and ideals in associative algebras 16P50 Localization and associative Noetherian rings 16W20 Automorphisms and endomorphisms Keywords:noetherian ring; prime radical; twisted polynomial rings; FBN ring; automorphism; endomorphisms; localization of twisted polynomial rings; finite normalizing extension; clans; PI ring PDFBibTeX XMLCite \textit{R. F. Damiano} and \textit{J. Shapiro}, J. Algebra 92, 116--127 (1985; Zbl 0558.16001) Full Text: DOI References: [1] Gauchon, G., Anneaux de polynomes essentiellement bornes, (Lecture Notes in Pure and Applied Mathematics, Vol. 51 (1979), Dekker: Dekker New York) · Zbl 0427.16011 [2] Chatters, A. W.; Heinicke, A. G., Localization at a torsion theory in hereditary noetherian rings, (Proc. London Math. Soc. (3), 27 (1973)), 193-204 · Zbl 0261.16001 [3] Damiano, R. F.; Papp, Z., On consequences of stabillity, Comm. Algebra, 9, 747-764 (1981) · Zbl 0463.16004 [4] Eisenbud, D.; Robson, J. C., Hereditary noetherian prime rings, J. Algebra, 16, 86-104 (1970) · Zbl 0211.05701 [5] Fields, K., On the global dimension of skew polynomial rings, J. Algebra, 13, 1-4 (1969) · Zbl 0181.04803 [6] Gordon, R., Polynomial modules over Macauley modules, Canad. Math. Bull., 19, 2, 173-176 (1976) · Zbl 0355.16018 [7] Gordon, R.; Robson, J. C., Krull dimension, Mem. Amer. Math. Soc., 133 (1973) · Zbl 0269.16017 [8] Heinicke, A. G.; Robson, J. C., Normalizing extensions prime ideals and incomparability, J. Algebra, 72, 237-268 (1981) · Zbl 0471.16018 [9] Jacobson, N., Structure of rings, (Colloquium Publications, Vol. 37 (1956), Amer. Math. Soc: Amer. Math. Soc Providence, R. I), (revised 1964) [10] Jategaonkar, A. V., Left principal ideal rings, (Lecture Notes in Mathematics No. 123 (1970), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0192.37901 [11] Malcolmson, P.; Shapiro, J., The Hilbert basis theorem for skew polynomial rings, Houston J. Math., 8, 3, 359-368 (1982) · Zbl 0507.16003 [12] Müeller, B. J., Localization in fully bounded noetherian rings, Pacific J. Math., 67, 233-245 (1976) [13] Rowen, L. H., Polynomial Identities in Ring Theory (1980), Academic Press: Academic Press New York · Zbl 0461.16001 [14] Shapiro, J., \(R\)-sequences in fully bounded noetherian rings, Comm. Algebra, 7, 819-831 (1979) · Zbl 0398.16001 [15] Stenström, B., Rings of Quotients (1975), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0296.16001 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.