The set of polynomials in one variable with coefficients in is the set of all formal linear combinations on elements , thought of as powers of the variable
where is an arbitrary natural number and , modulo the equivalence relation generated by equations of the form
(so that we ignore coefficients of zero).
This set is equipped with the structure of a ring itself, in fact a commutative algebra over , denoted and called the polynomial ring or ring of polynomials given by the unique bilinear map
which on monomials is given by