Suppose that
R is a (not necessarily
commutative)
ring, \sigma \colon R \to R is a
ring homomorphism, and \delta\colon R\to R is a '
σ
-derivation' of
R, which means that \delta is a
homomorphism of
abelian groups satisfying : \delta(r_1 r_2) = \sigma(r_1)\delta(r_2)+\delta(r_1)r_2. Then the
Ore extension R[x;\sigma,\delta], also called a
skew polynomial ring, is the
noncommutative ring obtained by giving the
ring of polynomials R[x] a new multiplication, subject to the identity : x r = \sigma(r)x + \delta(r). 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 ==