Ore algebra

In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. The concept is named after Øystein Ore.

Definition

Let $K$ be a (commutative) field and $A=K[x_{1},\ldots ,x_{s}]$ be a commutative polynomial ring (with $A=K$ when $s=0$). The iterated skew polynomial ring $A[\partial _{1};\sigma _{1},\delta _{1}]\cdots [\partial _{r};\sigma _{r},\delta _{r}]$ is called an Ore algebra when the $\sigma _{i}$ and $\delta _{j}$ commute for $i\neq j$, and satisfy $\sigma _{i}(\partial _{j})=\partial _{j}$, $\delta _{i}(\partial _{j})=0$ for $i>j$.

Properties

Ore algebras satisfy the Ore condition, and thus can be embedded in a (skew) field of fractions.

The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis for their left ideals.