Simple skew polynomial rings. (English) Zbl 0567.16003

Voskoglou, Michael G.

Publ. Inst. Math., Nouv. Sér. 37(51), 37-41 (1985).

We treat two questions. First we give general conditions for the existence of skew polynomial rings in finitely many variables over a given ring \(R\) (special cases of such rings are well known, typified by the \(n\)-th Weyl algebras) and second we obtain necessary and sufficient conditions for the simplicity of such rings. Our main result is as follows: Let \(R\) be a ring and let \(D=\{d_ 1,\ldots,d_ n\}\) be a set of derivations of \(R\) such that \(d_ i\circ d_ j=d_ j\circ d_ i\), for all \(i,j=1,\ldots,n\). Then if \(R\) is a \(D\)-simple ring of characteristic zero and d is an outer derivation of \(S_{i-1}\), for each \(i=1,\ldots,n\) (where \(S_ 0=R)\), the skew polynomial ring \(S_ n=R[x_ 1,d_ 1]\cdots[x_ n,d_ n]\) is simple. A similar argument holds if \(R\) is of prime characteristic, say \(p\). Note that Amitsur obtained conditions under which an Ore extension \(R[x,d]\) over a simple ring \(R\) is simple and more recently Jordan obtained such conditions if \(R\) is \(d\)-simple. These results can be reproduced as corollaries of our results.

Reviewer: Michael G. Voskoglou

MSC:

16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)
16D30	Infinite-dimensional simple rings (except as in 16Kxx)
16W20	Automorphisms and endomorphisms

Keywords:

skew polynomial rings in finitely many variables; n-th Weyl algebras; derivations; outer derivation; Ore extension; simple ring