Integral closure
Let be a ring and
a subring of
. We say that an element
is integral over
if there is a monic polynomial
, with each
so that
. The elements of
that are integral over
form a subring
of
which contains
. We call
the integral closure of
in
. If
, then we say that
is integral over
. If
, then we say that
is integrally closed in
. If
is integrally closed in its field of fractions, then we call it integrally closed.