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 formula_1 be a (commutative) field and formula_2 be a commutative polynomial ring (with formula_3 when formula_4). The iterated skew polynomial ring formula_5 is called an Ore algebra when the formula_6 and formula_7 commute for formula_8, and satisfy formula_9, formula_10 for formula_11.
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.