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MATH 340
Exam III – Part I (In Class)
Dr. Morton
Name:
Directions: You have 50 minutes in which to complete this exam. Please make sure that you read
through this entire exam before attempting any problems. You must show all work, or risk losing
credit. Be sure to answer all questions asked.
To receive full credit on problems, they must not only be mathematically correct, but they must
also be solved using the correct notation and terminology.
Good luck!
1. ( points) a. Describe all zero-divisors of
b. Describe all units of
§ ¢ .
§ ¢ .
c. Define an idempotent of a ring. Then describe all idempotents of
d. Define a nilpotent in a ring. Then describe all nilpotents of
e.  §  ¢ is a ring (you don’t have to check this). Is
Explain.
f.
 §  ¢ is a ring (you don’t have to check this). Is
§ ¢ .
§ ¢ .
 §  ¢ a commutative ring?
 §  ¢ a field? Explain.
  a b 



 0 0 

and
let
:
a,b,c,d
¢
S

:
c
¢


.



 c 0 

  c d 



a. Prove that S is a subring of M 2 (¢ ).
2. ( points) Let M 2 (¢ )   
b.
Is S a one- or two-sided ideal? Prove your point.
3. ( points) a. If a is a zero-divisor in a ring and n is a positive integer, prove that a n is either
zero or a zero-divisor.
b. Show that in a ring a non-zero element a is a zero-divisor if and only if aba=0 for some
b  0.
4. Let 2 [x] be the ring of all polynomials with coefficients in 2 (that is, coefficients are 0 or
1, and addition and multiplication of coefficients is done modulo 2). Recall that if R is a
ring and a is an element of R that a  {ra :r R} .
a. List the elements in
2
[x] / x 2  x  1 . (Hint: there are less than or equal to 5 different
cosets here.)
b. Make an addition and multiplication table for the cosets from part a.
c.
We showed in class that a =the principal ideal generated by a is an ideal of R. Thus
we know that
2
[x] / x 2  x  1 is a ring, by our theorem guaranteeing that R/A is a
ring iff A is an ideal of R. (All of this we proved together in class.)
Now prove, using part b that
2
[x] / x 2  x  1 is a field. Do NOT use any theorems
about prime or maximal ideals here.
5. ( points) a. (Finish the statement of )the Fundamental Theorem of Ring Homomorphisms:
Let  be a ring homomorphism from R to S, where R,S are rings. Then ….
b. Prove that

 ¢ /  a  b
 is ring-isomorphic to
Theorem of Ring Homomorphisms.
a
 ¢ b , using the Fundamental