The ideal extension of 
, an ideal
in a commutative ring 
, in a commutative ring 
with a multiplicative
identity, is the ideal generated by its image 
under a ring
homomorphism 
. Explicitly, it consists
of all finite sums of
the form 
where 
is in 
and 
is in 
. Sometimes the extension of 
is denoted 
.
The image 
may not be an ideal if 
is not a surjection. For instance,
![f:Z->Z[x]](/images/equations/IdealExtension/Inline15.svg)
is a ring
homomorphism and the image of the even integers is not an ideal since
it does not contain any nonconstant polynomials. The
extension of the even integers
in this case is the set of polynomials
with even coefficients.
The extension of a prime ideal may not be a prime ideal. For example, consider ![f:Z->Z[sqrt(2)]](/images/equations/IdealExtension/Inline16.svg)
. Then the extension
of the even integers
is not a prime ideal since 
while 
does not belong to the extended ideal 
.
