Ore extension

Ore extension

In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Oystein Ore, is a special type of a ring extension whose properties are relatively well understood. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.

Contents

Definition

Suppose that R is a ring, σ:R → R is an injective ring homomorphism, and δ:R → R is a σ-derivation of R, which means that δ is a homomorphism of abelian groups satisfying

δ(r1r2) = (σr1)δr2 + (δr1)r2.

Then the Ore extension R[λ;σ,δ] is the ring obtained by giving the ring of polynomials R[λ] a new multiplication, subject to the identity

λr = (σr)λ + δr.

If δ = 0 (i.e., is the zero map) then the Ore extension is denoted R[λ; σ] and is called a skew polynomial ring. If σ = 1 (i.e., the identity map) then the Ore extension is denoted R[λ,δ] and is called a differential polynomial ring.

Examples

The Weyl algebras are Ore extensions, with R any a commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative.

Properties

References

  • 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 0-521-83687-5; 0-521-54537-4, MR2080008 
  • 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, MR1811901 
  • Rowen, Louis H. (1988), Ring theory, vol. I, II, Pure and Applied Mathematics, 127, 128, Boston, MA: Academic Press, ISBN 978-0-12-599841-3, ISBN 0-12-599842-0, MR940245 
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