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, ${\displaystyle \sigma \colon R\to R}$ is a ring homomorphism, and ${\displaystyle \delta \colon R\to R}$ is a σ-derivation of R, which means that ${\displaystyle \delta }$ is a homomorphism of abelian groups satisfying

${\displaystyle \delta (r_{1}r_{2})=\sigma (r_{1})\delta (r_{2})+\delta (r_{1})r_{2}}$.

Then the Ore extension ${\displaystyle R[x;\sigma ,\delta ]}$, also called a skew polynomial ring, is the noncommutative ring obtained by giving the ring of polynomials ${\displaystyle R[x]}$ a new multiplication, subject to the identity

${\displaystyle xr=\sigma (r)x+\delta (r)}$.

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[1] (or invariant[2] ), if R·f = f·R, and
• central, if g·f = f·g for all g ∈ R.

Further reading

• Goodearl, K. R.; Warfield, R. B., Jr. (2004), An Introduction to Noncommutative Noetherian Rings, Second Edition, London Mathematical Society Student Texts, 61, Cambridge: Cambridge University Press, ISBN 0-521-54537-4, MR 2080008
• McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian rings, Graduate Studies in Mathematics, 30, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2169-5, MR 1811901
• Azeddine Ouarit (1992) Extensions de ore d'anneaux noetheriens á i.p, Comm. Algebra, 20 No 6,1819-1837. https://zbmath.org/?q=an:0754.16014
• Azeddine Ouarit (1994) A remark on the Jacobson property of PI Ore extensions. (Une remarque sur la propriété de Jacobson des extensions de Ore a I.P.) (French) Zbl 0819.16024. Arch. Math. 63, No.2, 136-139 (1994). https://zbmath.org/?q=an:00687054
• Rowen, Louis H. (1988), Ring theory, vol. I, II, Pure and Applied Mathematics, 127, 128, Boston, MA: Academic Press, ISBN 0-12-599841-4, MR 0940245

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.