Pairwise commuting derivations of polynomial rings

The Commuting Derivations Conjecture

This paper proves the Commuting Derivations Conjecture in dimension three: if D1 and D2 are two locally nilpotent derivations which are linearly independent and satisfy [D1, D2] = 0 then the intersection of the kernels, A1 ∩ A2 equals C[f ] where f is a coordinate. As a consequence, it is shown that p(X)Y + Q(X, Z, T ) is a coordinate if and only if Q(a, Z, T ) is a coordinate for every zero a ...

Global Dimension of Polynomial Rings in Partially Commuting Variables

For any free partially commutative monoid M(E, I), we compute the global dimension of the category of M(E, I)-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert’s Syzygy Theorem to polynomial rings in partially commuting variables.

Differential Polynomial Rings of Triangular Matrix Rings

Commuting $pi$-regular rings

R is called commuting regular ring (resp. semigroup) if for each x,y $in$ R there exists a $in$ R such that xy = yxayx. In this paper, we introduce the concept of commuting $pi$-regular rings (resp. semigroups) and study various properties of them.

Fields with several Commuting Derivations

The existentially closed models of the theory of fields (of arbitrary characteristic) with a given finite number of commuting derivations can be characterized geometrically, in several ways. In each case, the existentially closed models are those models that contain points of certain differential varieties, which are determined by certain ordinary varieties. How can we tell whether a given syst...