nLab Ore extension

Idea

An Ore extension of a unital ring $R$ is certain generalization of the ring $R[T]$ of polynomials in one variable $T$ with coefficients in $R$.

While keeping the left $R$-module-structure intact, unlike in the polynomial ring, the coefficients in $R$ and the indeterminate $T$ 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 $\sigma: R\to R$, a $\sigma$-derivation $d: R\to R$ is an additive map satisfying the $\sigma$-twisted Leibniz rule

$$d(r s) = d(r) s + \sigma(r) d(s),\,\,\,\,\,\forall r,s\in R.$$

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

$$T \cdot r = \sigma(r) T + d(r), \,\,\,\,\forall r\in R.$$

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

An Ore extension of an Ore extension of…(finitely many times) is called an iterated Ore extension. See quantum nilpotent algebra for an example.

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