Math 582 Assignment 2 Solutions
1. Let G be a nite group of order pqr, where p, q, r are primes with p < q < r. Prove
that G has a normal r-Sylow subgroup.
Solution. Let nr be the number of r-Sylow subgroups of G. Then nr is 1 modulo r
and divides pqr. Th
Math 582 Assignment 7 Solutions
1. Let V be a vector space over a eld F . Prove that V is Noetherian if and only if V is
Artinian, if and only if dim(V ) < .
Solution. Suppose that dim(V ) < , and let V1 V2 be an increasing chain
of subspaces. Then dim(V1
Math 582 Assignment 6 Solutions
Instructions. Throughout R will denote a commutative ring (with 1) and S will denote
a multiplicatively closed subset of R containing 1. If I, J are ideals of R, then (I : J ) :=
cfw_r R : rJ I is called the ideal quotient
Math 582 Assignment 5 Solutions
1. Let M, N be R-modules. Prove that M N is projective if and only if both M and
N are projective.
Solution. Suppose that M, N are projective. Let g : B C be a surjective Rmodule homomorphism and let : M N C be an R-module
Math 582 Assignment 3 Solutions
1. Let cfw_Mi : i I be a family of R-modules, and let P be an R-module.
(a) Prove that
homR
Mi , P
=
i I
homR (Mi , P )
i I
as Abelian groups.
(b) Prove that
homR
P,
Mi
i I
=
homR (P, Mi )
iI
as Abelian groups.
Solution. (
Math 582 Assignment 4 Solutions
Instructions. Unless otherwise specied, R denotes a commutative ring (with 1).
1. Let R be a commutative ring and let I, J be ideals of R. Prove that (R/I ) R (R/J )
=
R/(I + J ) as R-modules.
Solution. There is a homomorp
Math 582 Assignment 8 Solutions
1. Let A be a Noetherian commutative ring. If M, N are Noetherian A-modules, prove
that M A N is Noetherian.
Solution. Since A is Noetherian, it is sucient to prove that M A N is nitely
generated. Since M and N are Noetheri
Math 582 Exam 2
due Wednesday 24 April
Instructions. You may not consult anybody other than me on this exam, and you are
welcome to talk to me about it.
In Problem 4 below you are free to use the following fact without proof: If A B are
commutative rings
Math 582 Assignment 9 Solutions
Instructions. Throughout this assignment A will be a Dedekind domain.
1. If I and J are nonzero ideals of A, we say I divides J if there is an ideal K with
J = IK . Prove that I divides J if and only if J I .
Solution. If J
Math 582 Final Exam
8 May 2013
Instructions. Recall that if : A B is a ring homomorphism and P is an B -module,
then P is an A-module via r p = (r)p. If M is an A-module, then M A B is an B -module
via s (m t) = m st. More generally, if M is an A-module a
Math 582 Final Exam Solutions
Instructions. Recall that if : A B is a ring homomorphism and P is an B -module,
then P is an A-module via r p = (r)p. If M is an A-module, then M A B is an B -module
via s (m t) = m st. More generally, if M is an A-module an
Math 582 Exam 2 Solutions
In Problem 4 below you are free to use the following fact without proof: If A B are
commutative rings for which there is an ideal I of B with I A, then there is a 1-1 inclusion
preserving correspondence between the prime ideals o
Math 582 Exam 1 Solutions
1. Let I be a nonzero ideal of a commutative ring R. Prove that I is a free R-module if
and only if I = Ra for some a I which is not a zero divisor.
Solution. If I = Ra then there is a surjective homomorphism : R I dened by
(r) =
Math 582 Exam 1
due Friday 8 March
Instructions. You may not consult anybody other than me on this exam.
1. Let I be a nonzero ideal of a commutative ring R. Prove that I is a free R-module if
and only if I = Ra for some a I which is not a zero divisor.
2
Math 582 Assignment 1 Solutions
Throughout this assignment G refers to a nite Abelian group. Recall Problem 4 from
Assignment 4 of Math 581: If p is a divisor of |G|, set
Gp = cfw_g G : o(g ) is a power of p.
Then Gp is a subgroup of G. Furthermore, if |G