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, formula_1 is a ring homomorphism, and formula_2 is a "σ"-derivation of "R", which means that formula_3 is a homomorphism of abelian groups satisfying
Then the Ore extension formula_5, also called a skew polynomial ring, is the noncommutative ring obtained by giving the ring of polynomials formula_6 a new multiplication, subject to the identity
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.
Examples.
The Weyl algebras are Ore extensions, with "R" any 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.
Elements.
An element "f" of an Ore ring "R" is called