August 2008
Qualifying Examination
Algebra Part
There are only 6 questions. Do them all.
1. Let A be a nite abelian group. Prove that A is not a projective Z-module
and also that it is not an injective Z-module.
2. Prove that Q/Z Z Q/Z = 0
3. Let I be an
(Topics for qualifying exam in algebra) 731 SYLLABUS
I. Set-up (over rings with unity, including noncommutative)
Modules and Homomorphism Theorems
Direct Sums and Products, Free Modules
(including Universal Mapping Properties)
Projective and Injective Mod
Algebra Part of Qualifying Examination, August 23, 2010
Instructions: Do all questions, justify your answers with the necessary proofs. All
rings are associative (not necessarily commutative) with identity, and all modules are left
unitary modules. We den
Q ualifying Exam - January 2009
Algebra Part
Instructions: C omplete as m any q uestions a s possible. Answers should be justified with
the necessary proofs. All rings are a ssumed to be n oncom mutative unless stated
otherwise. All rings have an identity
Algebra P art o f Qualifying Examination, A ugust 25, 2009
Instructions: Do all questions, justify your answers with the necessary proofs.
All rings are associative (not necessarily commutative) with identity a nd all modules
are left unitary modules. We
Qualifying Examination
January 10, 2008
Algebra Part
Please do all ve questions.
Problem #5 is worth twice as much as each of the others
We will always assume that rings have an identity element and that modules
are unitary left modules.
1. Let I be an