Solutions to Homework 1.
All rings are commutative with identity!
(1) [4pts] Let R be a nite ring. Show that R = NZD(R).
Proof. Let a NZD(R) and ta : R R the map dened by ta (r) = ar for all
r R. Since a is a nonzero divisor on R, ta is injective. An inje
Solutions to Homework 2.
(1) [4pts] Let R be a ring and Q R an ideal with radQ = P where P R is a
prime ideal. Show that Q is P -primary if and only if for all a, b R with ab Q
and a P we have that b Q.
/
Proof. Suppose that Q is P -primary. By (2.21) for
Solutions to Homework 4.
(1) [10pts] Let R be a semilocal Noetherian ring and I R an ideal of R. Show
that the following conditions are equivalent:
(a) I is an ideal of denition of R.
(b) I Jrad(R) and R/I is an Artinian ring.
(c) I Jrad(R) and R/I has ni
Solutions to Homework 3.
(1) [6pts] For a polynomial P (t) Q show that the following conditions are equivalent:
(a) P (n) Z for all integers n Z.
(b) P (n) Z for all but nitely many integers n Z.
n
(c) P (t) = i=0 ai t with ai Z and n N suitable.
i
Proof.
Solutions to Homework 5.
(1) Prove the Five Lemma:
Consider a commutative diagram with exact rows:
f1
f2
f3
f4
A1 A2 A3 A4 A5
t
t
t
t
t
g1
g2
5
4
3
2
1
g3
g4
B1 B2 B3 B4 B5
and prove:
(a) [4pts] If t2 and t4 are surjective and t5 is injective, then t3 is