Editing Polynomial ring (section)

===Differential and skew-polynomial rings===
{{Main|Ore extension}}

Other generalizations of polynomials are differential and skew-polynomial rings.

A '''differential polynomial ring''' is a ring of [[differential operator]]s formed from a ring ''R'' and a [[Derivation (abstract algebra)|derivation]] ''δ'' of ''R'' into ''R''. This derivation operates on ''R'', and will be denoted ''X'', when viewed as an operator. The elements of ''R'' also operate on ''R'' by multiplication. The [[function composition|composition of operators]] is denoted as the usual multiplication. It follows that the relation {{nowrap|1=''δ''(''ab'') = ''aδ''(''b'') + ''δ''(''a'')''b''}} may be rewritten
as
: <math>X\cdot a = a\cdot X +\delta(a).</math>

This relation may be extended to define a skew multiplication between two polynomials in ''X'' with coefficients in ''R'', which make them a [[noncommutative ring]].

The standard example, called a [[Weyl algebra]], takes ''R'' to be a (usual) polynomial ring ''k''[''Y''&thinsp;], and ''δ'' to be the standard polynomial derivative <math>\tfrac{\partial}{\partial Y}</math>. Taking ''a'' = ''Y'' in the above relation, one gets the [[canonical commutation relation]], ''X''⋅''Y'' − ''Y''⋅''X'' = 1. Extending this relation by associativity and distributivity allows explicitly constructing the [[Weyl algebra]]. {{harv|Lam|2001|loc=§1,ex1.9}}.

The '''skew-polynomial ring''' is defined similarly for a ring ''R'' and a ring [[endomorphism]] ''f'' of ''R'', by extending the  multiplication from the relation ''X''⋅''r'' = ''f''(''r'')⋅''X'' to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism ''F'' from the monoid '''N''' of the positive integers into the [[endomorphism ring]] of ''R'', the formula ''X''<sup>''n''</sup>⋅''r'' = ''F''(''n'')(''r'')⋅''X''<sup>''n''</sup> allows constructing a skew-polynomial ring. {{harv|Lam|2001|loc=§1,ex 1.11}}  Skew polynomial rings are closely related to [[crossed product]] algebras.