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

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

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

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