an:01749333
Zbl 1010.16024
Jordan, David A.
Normal elements of degree one in Ore extensions
EN
Commun. Algebra 30, No. 2, 803-807 (2002).
00083462
2002
j
16S36 17B37
Ore extensions; normal elements; quantum matrices; domains
An element \(c\) of a ring \(R\) is `normal' if \(Rc=cR\). This note concerns normal elements \(c\) of the form \(dx+e\) (with \(d,e\in A\)) of Ore extensions \(R=A[x;\sigma,\delta]\), where \(A\) is a domain. In particular it is proved that \(d\) is normal in \(A\) and that if \(e\) is regular modulo \(Ad\) then \(R/Rc\) is a domain. The author indicates how this can be applied to show that \({\mathcal O}/{\mathcal O}\Delta\) is a domain, when \(\mathcal O\) is the coordinate ring of \(n\times n\) quantum matrices over a field \(F\) and \(\Delta\) is the quantum determinant. A similar conclusion can be deduced for the analogous factor of the multiparameter ring of quantum matrices. The lemma has found recent application in work of \textit{K. R. Goodearl} and \textit{T. H. Lenagan} [Duke Math. J. 103, No. 1, 165-190 (2000; Zbl 0958.16025)].
Kenneth A.Brown (Glasgow)
Zbl 0958.16025