MAT 612
03/05/10, due Friday 03/12/10
25 points
HW #4
Name
1. Prove: If P is a prime ideal, then P is an irreducible ideal.
2. Find a standard primary decomposition of the ideal I = (x2 , xy, 2) in the ring Z[x].
Note: It might be handy to use the followi
MAT 612
02/16/10, due Monday 02/22/10
25 points
HW #3
Name
1. Let R be a commutative ring with 1. Suppose f : M N is an R-module homomorphism, and K
be an R-module. Show that f induces a homomorphism f : HomR (N, K) HomR (M, K). Show
that (idM ) = idHomR
MAT 612
02/05/10, due Friday 02/12/10
25 points
HW #2
Name
1. Prove the ve lemma: suppose
p
q
r
s
V W X Y Z
V
p
q
r
s
W X Y
Z
is a commutative diagram of R-module homomorphisms with exact rows.
(a) Assume is onto and and are one-to-one. Show is one-to-o
MAT 612
HW #1
01/29/10, due Wednesday 02/03/10
25 points
1.
Name
(a) Prove: if R is left-noetherian, then Rn is noetherian as a left R-module. (This is Exercise
12.1.15 - refer to that exercise for a hint.)
(b) Show that a nitely-generated module over a n
MAT 612
Exam 2 (take-home)
04/07/2010 (due Wednesday 04/14/2010, 5 pm)
90 points
Name
1.(20) Let F E be a eld extension.
(a) Suppose E is transcendental over F. Prove that F [] is isomorphic to F [x]. (Hence F ()
is isomorphic to F (x).)
(b) A subset S =
MAT 612
Exam 1 (take-home)
2/22/2010 (due Monday 3/1/2010, 5 pm)
90 points
Name
All rings are commutative with 1. All modules are unital.
1.(20) (a) Suppose M and N are free R-modules of nite rank. Prove that HomR (M, N ) is a free
R-module, and determine
02/16/10, due Monday 02/22/10
25 points
1. Let R be a commutative ring with 1. Suppose f: M > N is an R-module homomorphism, and K
be an R-module. Show that f induces a homomorphism f*: HomR(N, K) % HomRMI, Show
that (idA/j)* : idHomR(M,K) and that 0 g
MAT 612
04/25/10
HW #6 Solutions1
9.3.1 Arguing by contradiction, suppose R is a UFD. Since p(x) = a(x)b(x) in F [x], and
p R[x], Gauss Lemma implies F such that a and 1 b are in R[x]. Since b is monic,
1 R, hence a = 1 (a) R[x], a contradiction. We dont
MAT 612
HW #5
03/24/10, due Wednesday 03/31/10
25 points
1.
Name
(a) A eld extension E F is algebraic if every element of E is algebraic over F. Prove that
E F is algebraic if and only if every element of E is contained in some eld K with
F K E and |K : F