Misplaced Pages

This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (November 2014) (Learn how and when to remove this message)

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Ore algebra" – news · newspapers · books · scholar · JSTOR (May 2024)

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 ${\displaystyle K}$ be a (commutative) field and ${\displaystyle A=K}$ be a commutative polynomial ring (with ${\displaystyle A=K}$ when ${\displaystyle s=0}$). The iterated skew polynomial ring ${\displaystyle A\cdots }$ is called an Ore algebra when the ${\displaystyle \sigma _{i}}$ and ${\displaystyle \delta _{j}}$ commute for ${\displaystyle i\neq j}$, and satisfy ${\displaystyle \sigma _{i}(\partial _{j})=\partial _{j}}$, ${\displaystyle \delta _{i}(\partial _{j})=0}$ for ${\displaystyle i>j}$.

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.

References

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

This algebra-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Computer algebra
• Ring theory
• Algebra stubs