polynomial ring,
n. 1. the ring, denoted R[X], of formal polynomials in X over a ring R, where X is a commuting indeterminate, that is, the ring of expressions of the form 
where aj is an element of R and n is a natural number. If 
and 
are members of the polynomial ring R[X], then their sum is
and their product is 
The definition can be extended inductively:
R[X1,..., Xn ] = (R[X1,..., Xn-1])[Xn],
where the Xi are distinct commuting indeterminates. Further, if R is a Noetherian ring with identity, then R[X1,..., Xn ] is Noetherian; this is the Hilbert basis theorem. In particular, if K is a field, then K[X1,..., Xn] is a Gaussian domain.
2. the ring, denoted R(X), of rational polynomials over X, that is, of ratios of elements of R[X].
where aj is an element of R and n is a natural number. If and are members of the polynomial ring R[X], then their sum isand their product is The definition can be extended inductively:R[X1,..., Xn ] = (R[X1,..., Xn-1])[Xn],
where the Xi are distinct commuting indeterminates. Further, if R is a Noetherian ring with identity, then R[X1,..., Xn ] is Noetherian; this is the Hilbert basis theorem. In particular, if K is a field, then K[X1,..., Xn] is a Gaussian domain.
2. the ring, denoted R(X), of rational polynomials over X, that is, of ratios of elements of R[X].