Ore algebra

Definition

Let K be a (commutative) field and A = K [ x 1 , … , x s ] be a commutative polynomial ring (with A = K when s = 0 ). The iterated skew polynomial ring A [ ∂ 1 ; σ 1 , δ 1 ] ⋯ [ ∂ r ; σ r , δ r ] is called an Ore algebra when the σ i and δ j commute for i ≠ j , and satisfy σ i ( ∂ j ) = ∂ j , δ i ( ∂ 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.