Structure of Rings of Differential Operators and Poisson Algebras

1. 设A是特征为O的域k上的仿射域，g是特征为m阶的A的k自同构。我们研究了A.D. bell引入的微分算子环D （A; g）。证明了如果A是A × g的固定子矩阵A‘上的自由模且基为1，则D （A; g）同构于D （A’）上的m × m矩阵环。由Grothendieck的一般平坦性定理可知，对于任意A， A的一个元素c使得D（A[1/ c]; g）同构于D（(A[1/ c])）上的m × m矩阵环。作为应用，我们确定了当A是k上的多项式环或洛朗多项式环，g是可对角线性自同构时D（A; g）的结构。如果环R的每一个（主）左理想的左湮灭子是由一个幂等子生成的，则称环R为（左主）拟贝尔。我们证明了如果R是（左主要）拟贝尔，G是一个有序单群，那么单群环RG又是（左主要）拟贝尔。当R是（左主要）拟baer， G是作用于R的有序群时，给出了偏群环r# G是（左主要）拟baer的充分必要条件。如果一个环R没有非零的幂零元素，我们就说它是约简的。设R为环，g为R的自同构，给出了偏多项式环R [x]的充分必要条件；我们还给出了R [x]的一些充分条件；g]到贝尔环、准贝尔环或主要准贝尔环。设B是Z- azumaya代数，D是导数，Z'是Z的元素Z使得D (Z)=0。证明了如果Z/Z′是指数1的纯不可分扩展，且B满足其他条件，则斜多项式环B [x；D]是Azumaya代数。设p为素数，G为p群，B为环，f为B的因子集。我们证明了如果叉积Δ （B， G, f）在B上是可分离的，并且如果p包含在B的Jacobson根中，那么Δ （B， G, f）/B是h可分离的扩展。引入了广义高阶导数的概念，并发展了它与普通高阶导数的关系。我们还研究了广义高阶导数的范畴性质。少

1. Let A be an affine domain over a field k of characteristic O, and g a k-automorphism of A of order m. We studied the ring D (A ; g) of differential operators introduced by A.D.Bell. We proved that if A is a free module over the fixed subring A' of A by g with a basis containing 1, then D (A ; g) is isomorphic to the ring of m by m matrices over D (A'). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c of A such that D(A[1/C] ; g) is isomorphic to the ring of m by m matrices over D ((A [1/c])). As an application, we determined the structure of D(A ; g) when A is a polynonmial or Laurent polynomial ring over k and g is a diagonalizable linear automorphism.2. A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. We showed that if R is (left principally) quasi-Baer and G is an ordered monoid, then the monoid ring RG is again (left principally) quasi- … More Baer. When R is (left principally) quasi-Baer and G is an ordered group acting on R, we gave a necessary and sufficient condition for the skew group ring R#G to be (left principally) quasi-Baer.3. A ring R is said to be reduced if it has no nonzero nilpotent elements. Let R be a ring and let g be an automorphism of R.We gave a necessaryand sufficient condition for the skew polynomial ring R [x ; g] to be reduced, We also gave some sufficient condition for R [x ; g] to a Baer ring, a quasi-Baer ring, or a principally quasi-Baer ring.4. Let B be a Z-Azumaya algebra, D a derivation, and Z' the element z of Z such that D (z)=0. We proved that if Z/Z' is a purely inseparable extension of exponent 1 and B satisfies some other conditions then the skew polynomial ring B [x ; D] is an Azumaya algebra. Let p be a prime, G a p-group, B a ring, and f a factor set of B.We showed tha if the crossed product Δ (B, G, f) is separable over B and if p is contained in the Jacobson radiucal of B then Δ (B, G, f)/B is an H-separable extension.5. We introduced the notion of generalized higher derivations and developed the relations between them and ordinary higher derivations. We also investigated the categorical properties of generalized higher derivations. Less