# nLab Ore extension

### Idea

An Ore extension of a unital ring $R$ is certain generalization of the ring $R\left[T\right]$ 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

$d\left(rs\right)=d\left(r\right)s+\sigma \left(r\right)d\left(s\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\forall r,s\in R.$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 $R\left[T\right]$ is equipped with the unique multiplication rule which is making it into a unital ring, extends $R=R1\subset R\left[T\right]$ and such that

$T\cdot r=\sigma \left(r\right)T+d\left(r\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\forall r\in R.$T \cdot r = \sigma(r) T + d(r), \,\,\,\,\forall r\in R.

$R\left[T\right]$ with this ring structure is called the Ore extension of $R$. If $d=0$ identically, then we say that $R\left[T\right]$ 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)