RELATIONS OF DERIVATIVE ALGEBRA AND RING OF DIFFERENTIAL OPERATORS IN CHARACTERISTIC p>0

Introduction　　 Let K be a field,K X ]∶=K x1,… ,xn]bethe polynomial ring in n variables.We know thatwhen K is a field with ch( K ) =0 ,then the Weylalgebra An( K) ,the ring of differential operatorsD( KX]) ,and the derivative algebraΔ ( K X])which is generated by{xi, i| i=1 ,… ,n}in End KKX]are all isomorphic1～ 3 ] .But if ch( K) =p>0 ,the three do not have that relation.In factΔ( KX]) is only a quotientof An( K) 4] ;and Ref.5 ]gives a comprehensive study to the rela…

RELATIONS OF DERIVATIVE ALGEBRA AND RING OF DIFFERENTIAL OPERATORS IN CHARACTERISTIC p>0

Let K be a field of characteristic p>0. We prove that the derivative algebra of Kx1,…,xn] is a proer subring of the ring of differential operators of Kx1,…,xn]. A concrete example is given to show that there is a differential operator of order p that does not belong to the derivative algebra. By these results, is follows that the derivative algebra is Morita equivalent to Kxp1,…,xpn], and hence its global homological dimension, Krull dimension, K0 group and some other properties are got.