An Ore extension uses a unital endomorphism of and a -derivation , satisfying . Its elements have unique expressions and multiplication is determined by the displayed relation. Applying the relation to shows why the endomorphism and derivation conditions are exactly the compatibility conditions for associativity. A differential Ore extension has . Adjoining ordinary partial derivatives successively to a polynomial ring gives the ordered-monomial construction of a Weyl algebra.
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The Ore extension, named after the mathematician Ole Johan Dahl Ore, is a concept in algebra that pertains to the extension of rings and modules. In particular, it is used to construct new rings from a given ring by adding new elements and defining new operations. The most common application of Ore extensions occurs in the context of noncommutative algebra, where it is used to form the Ore localization of a polynomial ring. This involves extending a ring by introducing new elements that satisfy specific relations concerning multiplication.