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Twisted polynomial rings satisfying a polynomial identity. (English) Zbl 0558.16001

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

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
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References:

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