nLab
Ore extension

Idea

An Ore extension of a unital ring RR is certain generalization of the ring R[T]R[T] of polynomials in one variable TT with coefficients in RR. While keeping the left RR-module structure intact, unlike in the polynomial ring, the coefficients in RR and the indeterminate TT do not need to commute, but rather commute up to a skew-derivation. A skew-polynomial ring is a special case.

Definition

Given an endomorphism σ:RR\sigma: R\to R, a σ\sigma-derivation d:RRd: R\to R is an additive map satisfying the σ\sigma-twisted Leibniz rule

d(rs)=d(r)s+σ(r)d(s),r,sR. d(r s) = d(r) s + \sigma(r) d(s),\,\,\,\,\,\forall r,s\in R.

If σ\sigma is an injective endomorphism of RR, and dd a σ\sigma-derivation dd then the free left RR-module underlying the ring of polynomials in one variable R[T]R[T] is equipped with the unique multiplication rule which is making it into a unital ring, extends R=R1R[T]R = R 1\subset R[T] and such that

Tr=σ(r)T+d(r),rR. T \cdot r = \sigma(r) T + d(r), \,\,\,\,\forall r\in R.

R[T]R[T] with this ring structure is called the Ore extension of RR. If d=0d = 0 identically, then we say that R[T]R[T] is a skew polynomial ring.

Literature

  • K. R. Goodearl, R. B. Warfield, An introduction to noncommutative Noetherian rings, London Math. Society Student Texts 61, Camb. Univ. Press.
  • Louis H. Rowen, Ring theory, student edition, Acad. Press 1991, sec. 1.6
  • wikipedia Ore extension

Created on September 15, 2011 19:20:06 by Zoran Škoda (161.53.130.104)