computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expression ...

, 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 Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with a ...

Definition

Let

K

be a (commutative) field and

A = K_1, \ldots, x_s /math> be a commutative polynomial ring (with A = K when s = 0). The iterated skew polynomial ring A partial_1; \sigma_1, \delta_1 \cdots partial_r; \sigma_r, \delta_r /math> is called an Ore algebra when the \sigma_i and \delta_j commute for i \neq j, and satisfy \sigma_i(\partial_j) = \partial_j, \delta_i(\partial_j) = 0 for i > j .

Properties

Ore algebras satisfy the

Ore condition In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, ...

, 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 In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Grö ...

for their left ideals.

References

Computer algebra Ring theory {{algebra-stub