Editing Polynomial ring (section)

===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]].