An Ore extension of a unital ring is certain generalization of the ring of polynomials in one variable with coefficients in . While keeping the left -module structure intact, unlike in the polynomial ring, the coefficients in and the indeterminate do not need to commute, but rather commute up to a skew-derivation. A skew-polynomial ring is a special case.
Given an endomorphism , a -derivation is an additive map satisfying the -twisted Leibniz rule
d(r s) = d(r) s + \sigma(r) d(s),\,\,\,\,\,\forall r,s\in R.
If is an injective endomorphism of , and a -derivation then the free left -module underlying the ring of polynomials in one variable is equipped with the unique multiplication rule which is making it into a unital ring, extends and such that
T \cdot r = \sigma(r) T + d(r), \,\,\,\,\forall r\in R.
with this ring structure is called the Ore extension of . If identically, then we say that is a skew polynomial ring.