An 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 
.
In the following adjoining-element formulas, suppose 
is a commutative ring
and 
is a subring with the same multiplicative
identity.
For every ring 
, the polynomial ring ![R[x]](/images/equations/ExtensionRing/Inline10.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 
.
For an extension 
of unit rings with the same multiplicative
identity, allowing noncommutative rings,
polynomial evaluation gives a ring homomorphism ![R[x]->S](/images/equations/ExtensionRing/Inline25.svg)
fixing 
only when the substituted element 
commutes with every element
of 
.
With several indeterminates, the substituted elements
must also commute with one another. Without these conditions, the generated subring
consists of sums of finite words in the adjoined elements
and coefficients from 
, rather than just ordinary polynomial expressions (Sharifi
n.d., §3.6).
