An integral element 
of 
over a commutative unit ring 
, where 
is an extension ring of
, is one of the roots of a monic polynomial with coefficients
in 
.
More generally, let 
be an identity-preserving ring homomorphism
between commutative rings. An element 
is integral over 
iff it satisfies a monic
polynomial relation
 |
for a positive integer 
and coefficients 
. This is equivalent to integrality over the
subring 
of 
, and does not require 
to be injective (Stacks Project,
Tag 00GH). For example, every element 
of 
is integral over 
under the natural ring homomorphism.
Choosing an integer 
mapping to 
gives the monic polynomial
relation 
.
