DEFINITION 07.2
Let
be a prime ideal of a ring
. Then we define its height
to e the supremum of the lengths of prime chaines

We have
.

DEFINITION 07.3
For an ideal
of a ring
, we define its height
to be

DEFINITION 07.4
Let
be a ring and
be an
-module. Then a prime ideal
of
is called an associated prime ideal of
if It is the annihilator ann(Mx
of some
.
The associated primes of the
-module
are refereedd to as the prime divisorsof
.

THEOREM 07.5Let
be a Noetherian ring, and
an ideal generated
by
elements; then if
is a minimal prime divisor of
we have
.