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 ![Z[i]](/images/equations/ExtensionRing/Inline5.svg)
are extension rings of the ring of integers 
.
For every ring 
,
the polynomial ring ![R[x]](/images/equations/ExtensionRing/Inline8.svg)
is a ring extension of 
. If 
is a ring extension of 
, and 
, the set
![R[a]={f(a)|f(x) in R[x]},](/images/equations/ExtensionRing/NumberedEquation1.svg) |
is the smallest subring of 
containing 
and 
, and is a ring extension of 
. More generally, given finitely many elements 
of 
, we can consider
![R[a_1,...,a_n]={f(a_1,...,a_n)|f(x_1...,x_n) in R[x_1,...,x_n]},](/images/equations/ExtensionRing/NumberedEquation2.svg) |
which is the ring extension of 
in 
generated by 
.
