skew polynomial ring

An ordinary polynomial ring R[x] lets x commute peacefully with all the coefficients: x*a = a*x. A skew polynomial ring keeps the same polynomials but rewires the rule for moving x past a coefficient, twisting it through an endomorphism of the base ring. Now x*a equals sigma(a)*x for some fixed endomorphism sigma, so dragging x to the left transforms each coefficient. It is the polynomial ring's noncommutative cousin, built to encode twists like shifts, Frobenius, or derivations.

Precisely, given a ring R and a ring endomorphism sigma of R, the skew polynomial ring R[x; sigma] consists of left polynomials a_0 + a_1*x + ... + a_n*x^n with the multiplication forced by the single rule x*a = sigma(a)*x for all a in R. More generally an Ore extension R[x; sigma, delta] also allows a sigma-derivation delta, giving the rule x*a = sigma(a)*x + delta(a), which simultaneously generalizes skew polynomials and differential operators.

These rings are a versatile factory of examples. With sigma the identity and delta an ordinary derivation, R[x; 1, delta] is a ring of differential operators — the Weyl algebra arises this way from k[t] with delta = d/dt. With delta = 0 and sigma a nontrivial automorphism, one gets the twisted shift algebras central to q-difference equations and quantum algebra. Many quantum planes and quantum groups are built from iterated Ore extensions.

They inherit good structure when the base is nice: if R is a division ring and sigma is an automorphism, R[x; sigma, delta] is a left and right principal ideal domain in which a noncommutative division algorithm holds, so one can do Euclidean-style arithmetic. This makes skew polynomial rings a controlled, computable playground where many phenomena of noncommutative algebra can be exhibited explicitly.