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 $K$ 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, 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

Computer algebra
Ring theory

{{algebra-stub