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Simple skew polynomial rings. (English) Zbl 0567.16003
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.

##### 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
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