A note on skew polynomial rings. (English) Zbl 0830.16019

Voskoglou, Michael G.

Publ. Inst. Math., Nouv. Sér. 55(69), 23-28 (1994).

The author considers the simplicity of a skew polynomial ring \(S_n = R[x_1,\dots, x_n; d_1, \dots, d_n]\) where \(R\) is a ring of prime characteristic \(p\) and \(d_1, \dots, d_n\) are commuting derivations of \(R\). He establishes a sufficient condition, involving the derivations of the intermediate rings \(S_{i-1}\) of the form \(\sum^m_{k = 0} c_k d^{p^k}_i\), where each \(c_k \in \bigcap^n_{j = k} \text{ker} (d_j)\), for the simplicity of \(S_n\) and proves the necessity of a weaker condition. As the author points out, D. R. Malm [Pac. J. Math 132, 85-112 (1988; Zbl 0608.16005)] has given a necessary and sufficient condition, in terms of the derivations of \(R\) of the form \(\sum^n_{i = 1} \sum^m_{k = 0} c_{ik} d^{p^k}_i\), where each \(c_{ik} \in \bigcap^n_{j=1} \text{ker} (d_j)\) and is central in \(R\), for the simplicity of \(S_n\).

Reviewer: D.A.Jordan (Sheffield)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16D25	Ideals in associative algebras
16W25	Derivations, actions of Lie algebras
16D60	Simple and semisimple modules, primitive rings and ideals in associative algebras

Keywords:

skew polynomial ring; commuting derivations; intermediate rings; simplicity

Citations:

Zbl 0636.16002; Zbl 0608.16005