Example

Let R be a ring and take S to be the polynomial ring R[X]. Then the polynomial ring

S[Y]

in the indeterminate Y is the same as the ring of polynomials in the two indeterminates X, Y.

So its elements are of the form

_{i,j N} a_i,jXⁱY^j,

with a_i,j R. The element XY is equal to the product YX.

Thus, there is the notion of a polynomial ring in two indeterminates, X, Y, and the ring is isomorphic to (R[Y])[X]. It is also denoted by R[X,Y].