In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.
If δ = 0 (i.e., is the zero map) then the Ore extension is denoted R[x; σ]. If σ = 1 (i.e., the identity map) then the Ore extension is denoted R[x,δ] and is called a differential polynomial ring.
The Weyl algebras are Ore extensions, with R any a commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.
- An Ore extension of a domain is a domain.
- An Ore extension of a skew field is a non-commutative Principal ideal domain.
- If σ is an automorphism and R is a left Noetherian ring then the Ore extension R[λ;σ,δ] is also left Noetherian.
An element f of an Ore ring R is called
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