Given a commutative ring , the set of polynomials in one variable with coefficients in consists of all formal expressions
where is an arbitrary natural number and , modulo the equivalence relation generated by equations of the form
(so that we ignore coefficients of zero).
These polynomials also form a commutative ring, denoted , which is in fact a commutative algebra over ; the addition and multiplication in this ring are defined in the obvious way. The ring may also be described more abstractly as the free -algebra on one generator. Similarly, the set of polynomials in any give set of variables with coefficients in is the free commutative -algebra on that set of generators; see symmetric power and symmetric algebra.
The field of fractions of is the field of rational functions.