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.^[1] The concept is named after Øystein Ore.

Definition

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

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.