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.
Let be a (commutative) field and be a commutative polynomial ring (with when ). The iterated skew polynomial ring is called an Ore algebra when the and commute for , and satisfy , for .
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.