# 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.[1] The concept is named after Øystein Ore.

## Definition

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 .

## 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.

## References

- Chyzak, Frédéric; Salvy, Bruno (1998). "Non-commutative Elimination in Ore Algebras Proves Multivariate Identities".
*Journal of Symbolic Computation*. Elsevier.**26**(2): 187–227. doi:10.1006/jsco.1998.0207.

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