Math 905
Solutions to Homework # 7
Throughout R denotes a commutative ring with identity.
1. Let R be a Noetherian domain. Prove that R is a UFD if and only if every height one
prime is principal.
Solution: Recall that a Noetherian domain is a UFD if and
Math 905
Solutions to Homework # 6
Throughout R denotes a commutative ring with identity.
1. Let R be a Noetherian ring and I and J two ideals of R. Prove that I = J if and only if
Ip = Jp for all p AssR (R/I )
AssR (R/J ).
Solution: If I = J then certain
Jason Hardin
Math 905 - Homework 5
1. Let R be a Noetherian ring. Prove that R is reduced if and only if Rp is a eld for all p AssR R.
Proof. First suppose R is reduced. This means that (0) =
(0) =
p = p1 p2 ps ,
pSpecR
where p1 , . . . , ps are the disti
Math 905
Solutions to Homework # 4
Throughout R denotes a commutative ring with identity.
1. Prove that if R is Noetherian then so is R[x].
Solution: By Exercise 6 from Homework # 2, it suces to prove that every prime ideal
of R[x] is nitely generated. Le
Math 905
Solutions to Homework # 3
Throughout R denotes a commutative ring with identity.
1. Prove the following statements are equivalent:
(a) R is reduced
(b) Rp is reduced for all prime ideals p.
Also, prove that if R is reduced then Rp is a eld for al
Math 905
Solutions to Homework # 2
Throughout R denotes a commutative ring with identity.
1. Prove that a ring R is quasi-local if and only if the non-units of R form an ideal.
Solution: Suppose R is quasi-local with maximal ideal m. Then clearly m consis
Math 905
Solutions to Homework # 1
Throughout R denotes a commutative ring with identity.
1. Let I be an ideal of R. A prime ideal p containing I is said to be minimal over I if there
are no prime ideals properly contained in p which also contain I . Prov