Structure of Rings of Differential Operators and Poisson Algebras

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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