Editing Polynomial ring (section)

==Generalizations==

Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, [[noncommutative polynomial ring]]s, [[skew polynomial ring]]s, and polynomial [[Rig (mathematics)|rig]]s.

=== Infinitely many variables ===
One slight generalization of polynomial rings is to allow for infinitely many indeterminates.  Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials.  Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the [[free commutative algebra]], the only difference is that it is a [[free object]] over an infinite set.

One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the [[power series ring#Power series in several variables|ring of power series in infinitely many variables]]. Such a ring is used for constructing the [[ring of symmetric functions]] over an infinite set.

===Generalized exponents===
{{Main|Monoid ring}}
A simple generalization only changes the set from which the exponents on the variable are drawn.  The formulas for addition and multiplication make sense as long as one can add exponents: {{nowrap|1=''X''{{i sup|''i''}} ⋅ ''X''{{i sup|''j''}} = ''X''{{i sup|''i''+''j''}}}}.  A set for which addition makes sense (is closed and associative) is called a [[monoid]].  The set of functions from a monoid ''N'' to a ring ''R'' which are nonzero at only finitely many places can be given the structure of a ring known as ''R''[''N''], the '''monoid ring''' of ''N'' with coefficients in ''R''.  The addition is defined component-wise, so that if {{nowrap|1=''c'' = ''a'' + ''b''}}, then {{nowrap|1=''c''<sub>''n''</sub> = ''a''<sub>''n''</sub> + ''b''<sub>''n''</sub>}} for every ''n'' in ''N''.  The multiplication is defined as the Cauchy product, so that if {{nowrap|1=''c'' = ''a'' ⋅ ''b''}}, then for each ''n'' in ''N'', ''c''<sub>''n''</sub> is the sum of all ''a''<sub>''i''</sub>''b''<sub>''j''</sub> where ''i'', ''j'' range over all pairs of elements of ''N'' which sum to ''n''.

When ''N'' is commutative, it is convenient to denote the function ''a'' in ''R''[''N''] as the formal sum:
:<math>\sum_{n \in N} a_n X^n</math>

and then the formulas for addition and multiplication are the familiar:
:<math>\left(\sum_{n \in N} a_n X^n\right) + \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} \left(a_n + b_n\right)X^n</math>

and
:<math>\left(\sum_{n \in N} a_n X^n\right) \cdot \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} \left( \sum_{i+j=n} a_i b_j\right)X^n</math>

where the latter sum is taken over all ''i'', ''j'' in ''N'' that sum to ''n''.

Some authors such as {{harv|Lang|2002|loc=II,§3}} go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where ''N'' is the monoid of non-negative integers.  Polynomials in several variables simply take ''N'' to be the direct product of several copies of the monoid of non-negative integers.<!-- Quite tempting to say, ''N'' = '''N'''<sup>''n''</sup>. -->

Several interesting examples of rings and groups are formed by taking ''N'' to be the additive monoid of non-negative rational numbers, {{harv|Osborne|2000|loc=§4.4}}. See also [[Puiseux series]].

===Power series===
{{Main|Formal power series}}

Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms.  This requires various hypotheses on the monoid ''N'' used for the exponents, to ensure that the sums in the Cauchy product are finite sums.  Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums.  For the standard choice of ''N'', the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from ''N'' to a ring ''R'' with addition component-wise, and multiplication given by the Cauchy product.  The ring of power series can also be seen as the [[Completion of a ring|ring completion]] of the polynomial ring with respect to the ideal generated by {{mvar|x}}.

===Noncommutative polynomial rings===
{{Main|Free algebra}}

For polynomial rings of more than one variable, the products ''X''⋅''Y'' and ''Y''⋅''X'' are simply defined to be equal.  A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained.  Formally, the polynomial ring in ''n'' noncommuting variables with coefficients in the ring ''R'' is the [[monoid ring]] ''R''[''N''], where the monoid ''N'' is the [[free monoid]] on ''n'' letters, also known as the set of all strings over an alphabet of ''n'' symbols, with multiplication given by concatenation.  Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.

Just as the polynomial ring in ''n'' variables with coefficients in the commutative ring ''R'' is the free commutative ''R''-algebra of rank ''n'', the noncommutative polynomial ring in ''n'' variables with coefficients in the commutative ring ''R'' is the free associative, unital ''R''-algebra on ''n'' generators, which is noncommutative when ''n''&nbsp;>&nbsp;1.

===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.

=== Polynomial rigs ===
{{See also|Formal power series#On a semiring}}
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure ''R'' be a [[Field (mathematics)|field]] or a [[Ring (mathematics)|ring]] to the requirement that ''R'' only be a [[semifield]] or [[Rig (mathematics)|rig]]; the resulting polynomial structure/extension ''R''[''X''] is a '''polynomial rig'''. For example, the set of all multivariate polynomials with [[natural number]] coefficients is a polynomial rig.