Ore extension
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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 group algebras of polycyclic groups, enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.
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[edit] 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 if σ = 1 (i.e., the identity map) then the Ore extension is denoted R[λ,δ] and is called a differential polynomial ring.
[edit] Examples
The Weyl algebras are Ore extensions, with R any a commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative.
[edit] Properties
- An Ore extension of a domain is a 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.
[edit] References
- Goodearl, K. R. & Warfield, R. B., Jr. (2004), An Introduction to Noncommutative Noetherian Rings, Second Edition, vol. 61, London Mathematical Society Student Texts, Cambridge: Cambridge University Press, MR2080008, ISBN 0-521-83687-5; 0-521-54537-4
- McConnell, J. C. & Robson, J. C. (2001), Noncommutative Noetherian rings, vol. 30, Graduate Studies in Mathematics, Providence, R.I.: American Mathematical Society, MR1811901, ISBN 978-0-8218-2169-5
- Rowen, Louis H. (1988), Ring theory, vol. I, II, vol. 127, 128, Pure and Applied Mathematics, Boston, MA: Academic Press, MR940245, ISBN 978-0-12-599841-3, ISBN 0-12-599842-0
