If 
and 
are commutative unit rings,
and 
is a subring of 
, then 
is called integrally closed in 
if every element of 
which is integral over 
belongs to 
; in other words, there is no proper integral extension of
contained in 
.
If 
is an integral domain, then 
is called an integrally closed domain if it is integrally
closed in its field of fractions.
Every unique factorization domain is an integrally closed domain; e.g., the ring of integers 
and every polynomial
ring over a field are integrally closed domains.
Being integrally closed is a local property, i.e., every localization of an integrally closed domain is again an integrally closed domain.
