A extension ring (or ring extension) of a ring is any ring of which is a subring. For example, the field of rational numbers and the ring of Gaussian integers are extension rings of the ring of integers .
For every ring , the polynomial ring is a ring extension of . If is a ring extension of , and , the set
is the smallest subring of containing and , and is a ring extension of . More generally, given finitely many elements of , we can consider
which is the ring extension of in generated by .