## Normal elements of degree one in Ore extensions.(English)Zbl 1010.16024

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 K. R. Goodearl and T. H. Lenagan [Duke Math. J. 103, No. 1, 165-190 (2000; Zbl 0958.16025)].

### MSC:

 16S36 Ordinary and skew polynomial rings and semigroup rings 17B37 Quantum groups (quantized enveloping algebras) and related deformations

### Keywords:

Ore extensions; normal elements; quantum matrices; domains

Zbl 0958.16025
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