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.