Linear Groups and Projective Modules over Skew Polynomial Rings

d-Simple rings and simple -modules

“…It is easy to see that if such a derivation exists then Ω 1 (X) has a free direct summand. For example, if X is a surface in A 2 (K), the d-simplicity of O(X) implies that Ω 1 (X) is free; see [Arc,theorem 2•5•18]. No easy necessary and sufficient condition is known for checking that O(X) is d-simple for a given X.…”

“…This proposition is due to Shamsuddin; see [Arc,theorem 2•3•16]. It is the key to the construction of an important family of derivations of K [x] with respect to which this ring is d-simple.…”

“…1 )∂ 1 − 2x 1 ∂ 2 + 2x 3 1 ∂ 3 is a derivation of O(X). The following result comes from [Arc,theorem 2•5•23], but we give a more direct proof.…”

On cyclic essential extensions of simple modules over differential operator rings

“…The following lemma is known in the literature and its proof can be obtained in [1,Theorem 2.3.8] or in [10,Corollary 2.12]. We will denote by Max(R) the maximal spectrum of a ring R. The next result characterize differential operator rings R[θ; δ] satisfying (⋄) when R is an affine K-algebra which is an integral domain of Krull dimension 1.…”

On the Ideals of a Noetherian Ring

“…This appears in [1], but since it has not been published, we will outline the proof. The only nontrivial part of the proof is to show that S is simple, for which it suffices to show that no proper ideal of C is left invariant by 8.…”