Math 902
Solutions to Homework # 6
Throughout R denotes a ring with identity and F a eld.
1. Let R be a nite-dimensional semisimple F -algebra and M a simple left R-module.
Suppose char F does not divide dimF EndR M . Prove that M = 0.
Solution: (Note: Th
Math 902
Solutions to Homework # 5
Most of these solutions are due to Peder Thompson (with some help from others).
1. Let F be a eld, an innite set, R = i F , and I = i F . Prove that R I is not
isomorphic to a product of elds. (There are lots of ways to
Solutions to Homework #3
These solutions are due to Philip Gipson.
Problem 1. Suppose that I R is a simple left ideal. Prove that a) given y R that either Iy = 0
or Iy I , b) B (I ) = J I J is a two-sided ideal of R, and c) that if R is simple then in fac
Math 902
Solutions to Homework # 2
1. Prove that R is a division ring if and only if R has only two left ideals.
Solution: The hypothesis gives us that Rx = R for every nonzero x R. Hence,
every nonzero element has a left inverse. Let x be a nonzero eleme
Math 902
Solutions to Homework # 1
Most of these solutions are due to Nora Youngs (with a few edits here and there).
(1) Let S =
RR
. Prove that S is left Artinian but not right Artinian.
0Q
Proof. (This part of the solution is due to Michael Brown.) Let
Math 902
Final Exam
Monday, April 30th, 10:00am - 12:00 noon
Instructions: All problems are worth 10 points. You must work alone, but you may use your
notes and any other non-human resources (books, websites, etc) you wish. The problems can
all be solved
Math 902
Exam II
Due Date: 11:30am, Monday, April 16th
Instructions: Do ve of the seven problems below but no more than three from any one section.
All problems are worth 20 points. You must work alone, but you may use your notes and any
other non-human r
Math 902
Exam # 1
Instructions: Do ve of the seven problems below. All problems are worth 20 points.
1. Give an example of each of the following. No justication is needed.
(a) A commutative Artinian ring which is not semisimple.
(b) A ring containing an e
Math 902
Exam # 2
Instructions: Do ve of the seven problems below. All problems are worth 20 points.
1. Give an example of each of the following. No justication is needed.
(a) A noncommutative semisimple C-algebra.
(b) A simple ring which is not semisimpl