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Transcript
```MTH 200 – Graded Assignment 6
Material from Module 10 (Rings and Fields).
P1: Perform the arithmetic in the various Zm below.
a) 13  13
b) 7·5  4 in Z11
P2: Prove that multiplication in Zm , defined by
[a]m ·[b]m  [a·b]m
is well defined. Hints/outline in suggested problems somewhere in this module!
P3: For addition in Zm , defined by [a]m  [b]m  [a  b]m , prove the commutative
P4: Solve the given equations in the indicated Zm . Perform algebra using additive
inverses (always possible), and multiplicative inverses (whenever possible). Show that
you are using inverses explicitly!
If it is not possible to use a multiplicative inverse, switch to congruence notation and
apply the results for solving linear congruences. If no solution exists, be sure to indicate
this and state why. If you do not already have Cayley tables constructed for a particular
m , you may wish to construct some quickly for this problem to obtain inverses.
a) 2x  4  3 in Z5
b) 2x  4  3 in Z6
c) 5x  7 in Z13
d) 3x  6 in Z12
P5: Let R be the subset of
defined by
R  {x | x  a 3  7b, a, b  }
with the usual operations of real addition and multiplication defined on it.
R with those operations is a subring of . You don't have to verify all the properties
(you can take my word that it is), but do show that
a) R is closed under multiplication.
b) The multiplicative identity for is an element of R (can be expressed in the form
a 3  7b for some a, b  , and thus R is a ring with unity.
c) Bonus question: is R a proper subring (analogous to proper subset) of
in fact itself? Explain.
, or is it
P6: The sets R with Cayley table operations defined below both fail to be rings. Quickly
identify (at least) one property that each of them fails on. You can use the defining ring
properties, or the extended ones from the proposition in the "More Properties" section.
a) R  {a, b, c} with
b) R  {a, b, c, d } with

a
b
c
d
 a b c
* a b
c
a b c a
a a b
c
b
c a b
b b
c
a
c
a b c
c
c
a b
a b
c
d

a
b
c
d
b
c
d
a
a
a
c
b
d
c
a
b
d
b
c
c
b
d
a b
c
d
c
a
b
c
d
c
a
c
d
d
d
d
d
b
P7: Illustrate the distributive property of matrix multiplication over addition: let
1 2
4 0
 6 1
A
B
C



5 7 
0 2
 1 1
and show that A( B  C )  AB  AC .
P8: Verify that the ring property ( x) *( y )  x * y holds specifically for each element in
the Cayley table defined ring from the example. Use a truth table style setup for all
combinations, as in the suggested problems.
x
y
a
a
a
b
a
c
b
a
b
b
b
c
c
a
c
b
c
c
 a b c
* a b c
a
a
c a b
c b a
b a b c
b b b b
c
c
b c a
x* y
x
a b c
y
( x) *(  y )
P9:
 1 3
 2 1
a) Show that the matrices A  
and B  

 are zero divisors in M 2,2 .
 2 6 
 2 1
2
3
 1 2
b) Show that the matrices A  
and
B



 1 4 
1
 6
in M 2,2 .
1
 
3
 are multiplicative inverses
1 
6 
P10:
a) List the elements of Z 27 which are zero divisors. List which are units.
b) Give the multiplicative inverses for all elements which are units in Z12 .
P11: Prove that in any Z m , m 1 is always its own multiplicative inverse; i.e the product
(m  1)(m  1) is always equal to 1 in Z m .
P12: Prove that a ring has at most one unity (i.e. if a ring has a multiplicative identity,
then that identity must be unique).
P13: Give an example from a noncommutative ring that shows that ( x  y)2 is not
necessarily equal to x 2  2xy  y 2 . What is the correct expression for ( x  y)2 ? Hints 2.5
textbook exercises.
P14: For the complex numbers z  2  4i and w  3  3i find/compute
a) | z |
b) ( z )( w)
c)
z
w
d) arg( w)
e) w5
P15: Let z and w be complex numbers. Prove that z w  ( zw) .
P16: Find all the eight roots of unity (solutions to x8  1 in the complex plane). Work in
polar (using deMoivre's Theorem), then convert answers back to a  bi form using
r cos   (r sin  )i .
```
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