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 [math]\displaystyle{ K }[/math] be a (commutative) field and [math]\displaystyle{ A = K[x_1, \ldots, x_s] }[/math] be a commutative polynomial ring (with [math]\displaystyle{ A = K }[/math] when [math]\displaystyle{ s = 0 }[/math]). The iterated skew polynomial ring [math]\displaystyle{ A[\partial_1; \sigma_1, \delta_1] \cdots [\partial_r; \sigma_r, \delta_r] }[/math] is called an Ore algebra when the [math]\displaystyle{ \sigma_i }[/math] and [math]\displaystyle{ \delta_j }[/math] commute for [math]\displaystyle{ i \neq j }[/math], and satisfy [math]\displaystyle{ \sigma_i(\partial_j) = \partial_j }[/math], [math]\displaystyle{ \delta_i(\partial_j) = 0 }[/math] for [math]\displaystyle{ i \gt j }[/math].

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.