an:00759549
Zbl 0830.16019
Voskoglou, Michael G.
A note on skew polynomial rings
EN
Publ. Inst. Math., Nouv. Sér. 55(69), 23-28 (1994).
0350-1302
1994
j
16S36 16D25 16W25 16D60
skew polynomial ring; commuting derivations; intermediate rings; simplicity
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, \textit{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 \(S_n\).
D.A.Jordan (Sheffield)
0636.16002; 0608.16005