Ore extension

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. Elements of a Ore extension are called Ore polynomials. 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. Definition Suppose that R is a (not necessarily commutative) ring, is a ring homomorphism, and is a σ-derivation of R, which means that is a homomorphism of abelian groups satisfying