Ore extension

Definition

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

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.

Properties

• 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.

Elements

An element f of an Ore ring R is called

• twosided (or invariant ), if R·f = f·R, and
• central, if g·f = f·g for all g in R.

References

References

1. Jacobson, Nathan. (1996). "Finite-Dimensional Division Algebras over Fields". Springer.
2. Cohn, Paul M.. (1995). "Skew Fields: Theory of General Division Rings". [[Cambridge University Press]].