Let ${\displaystyle K}$  be a (commutative) field and ${\displaystyle A=K[x_{1},\ldots ,x_{s}]}$  be a commutative polynomial ring (with ${\displaystyle A=K}$  when ${\displaystyle s=0}$ ). The iterated skew polynomial ring ${\displaystyle A[\partial _{1};\sigma _{1},\delta _{1}]\cdots [\partial _{r};\sigma _{r},\delta _{r}]}$  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 ${\displaystyle i>j}$ .

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.