Math 724
Fall 2001
Homework 1
Due
September 17, 2001.
- Let
be a ring and 
an ideal of 
We define the radical
of I to be 
a)
Show
the radical of 
is itself an ideal.
b)
Show
that 
.
c)
Use
the previous parts to characterize the set of nilpotent elements of 
- Let
be a ring. Show
that if 
has an ideal
that is not finitely generated, then 
has a prime ideal
that is not finitely generated. Use this to show that every ideal of 
is finitely generated if and only if every prime ideal
of 
is finitely generated.
- (The Five Lemma)
Consider the following commutative diagram of
module homomorphisms with
exact rows:
Show that:
a)
If
is onto and 
and 
are one to one, then 
is one to one.
b)
If
is one to one and 
and 
are onto, then 
is onto
c)
Show
that this result has the short-five lemma as a corollary.