Math 902
Exam I
Due: 11:30 am on Wednesday, March 16th
Instructions: Do at most 7 problems. All problems are worth 20 points. You must work alone,
but you may use your notes and any other non-human resources (books, websites, etc) you wish.
However, the problems can all be solved using just results we have discussed in class or in the
homework, so your solutions should be confined to that material. (In other words, don’t use
theorems we haven’t proved in class.)
Except where specified, R denotes a commutative ring with identity.
1. Assume R is Noetherian and let M be a nonzero R-module. Let Λ be the set of all ideals
of the form (0 :R x) for some nonzero x ∈ M . Note that Λ is nonempty as M 6= 0. As
R is Noetherian (not by Zorn’s lemma!), every ideal in Λ is contained in some maximal
element of Λ. Prove that each maximal element of Λ is a prime ideal.
2. Let R be Noetherian and M an R-module. A prime ideal of R of the form (0 :R x) for
some x ∈ M is called an associated prime of M . The set of all associated primes of M
is denoted AssR M . By Problem # 1, M 6= 0 if and only if AssR M 6= ∅. Prove that the
following are equivalent for a prime ideal p:
(a) p ∈ AssR M .
(b) There exists an injective R-homomorphism R/p → M .
(c) HomRp (k(p), Mp ) 6= 0, where k(p) = Rp /pRp .
3. Let R be Noetherian and M a nonzero R-module. Let
ZDR (M ) = {r ∈ R | ru = 0 for some u ∈ M, u 6= 0}.
That is, ZDR (M ) is the set of elements of R which are zero-divisors on M . Prove that
[
ZDR (M ) =
p.
p∈AssR M
4. Let R be Noetherian and M and N R-modules. Assume M is finitely generated. Prove
that
AssR HomR (M, N ) = SuppR M ∩ AssR N.
(Hint: Use Problem # 2 and Hom-tensor adjunction. Also note that for a finitely generated
R-module L, L ⊗R k(p) ∼
= k(p)` , where ` = µRp (Lp ) by Nakayama.)
5. Let φ : R → S be a homomorphism of commutative rings. Let F be a flat S-module and
E an injective R-module. Prove that HomR (F, E) is an injective S-module.
6. Let M be a finitely presented R-module and {Ai | i ∈ I} a set of R-modules. Prove that
there is an isomorphism
Y
Y
∼
=
ψ : M ⊗R
Ai −
→
(M ⊗R Ai )
i
i
such that ψ(m ⊗ (ai )) = (m ⊗ ai ) for m ∈ M, ai ∈ Ai . (Hint: Use the right exactness of
the tensor product and a finite presentation for M .)
7. Let C[0, 1] denote the set of continuous functions f : [0, 1] → R, where [0, 1] denotes the
closed interval of real numbers from 0 to 1. It is easily seen that C[0, 1] is a commutative
ring under addition and multiplication (not composition!) of real-valued functions. (You
do not need to show this.)
(a) Describe the units of C[0, 1].
(b) Prove that Spec C[0, 1] is connected.
(c) For a ∈ [0, 1], show that ma := {f ∈ C[0, 1] | f (a) = 0} is a maximal ideal of C[0, 1].
8. Prove that every maximal ideal of C[0, 1] has the form ma for some a ∈ [0, 1], as defined
in the previous problem. (Hint: Suppose I is an ideal not contained in any ma . Use the
compactness of [0, 1] to show I contains a unit. Let a ∈ [0, 1]. Then there exists fa ∈ I
such that fa (a) 6= 0. Since fa is continuous, there exists an open set Oa containing a such
that fa (u) 6= 0 for all u ∈ Oa .)
Q
9. Let F be field and R = ∞
i=1 F , the product of countably many copies of F . Recall that
R is von Neumann regular. Let I = ⊕∞
i=1 F . Observe that I is an ideal of R. Prove that Ip
is an injective Rp -module for every prime p of R, but that I is not an injective R-module.
(Hint: Note that every nonzero ideal of R has nontrivial intersection with I.)
10. Let M be a finitely generated R-module and let µR (M ) denote the least integer ` such
that M has a generating set consisting of ` elements. Note that for any multiplicatively
closed set, µRS (MS ) ≤ µR (M ). For any integer r ≥ 0, prove that the set
Ur := {p ∈ Spec R | µRp (Mp ) ≤ r}
is an open set of Spec R. (Hint: Suppose p ∈ Ur . Prove there exists c ∈ R such that
p ∈ D(c) ⊆ Ur . Recall that D(c) = {q ∈ Spec R | c 6∈ q}.)