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 $A=K[x_1,\dots ,x_s]$ 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 $i\ne j$, 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.