Math 361 Test Question Prep
Jenny
1. True or False? Zmodn will always form a ring or a field but never both.
False - All fields are also rings, but not all rings are fields.
2. Form a non-trivial ideal from Zmod12 and prove that it is in fact an ideal.
a. show I is a subset of Zmod12
b. Show that I is a normal subgroup of Zmod12. It is normal because Zmod12 forms an abelian group. Subset is already proved from above, then
prove closure, and inverse.
c. Prove that I "attracts" products with Zmod12.
3. Consider Zmod20. Create four non-trivial ideals and circle the smallest generator of each set. What is the realtionship between the smallest
generator and the order of the ideal it lives in?
{0,10} - Smallest generator: 10 Order: 2
{0,2,4,6,8,10,12,14,16,18} - Smallest generator: 2 Order: 10
{0,4,8,12,16} - Smallest generator: 4 Order: 5
{0,5,10,15} - Smallest generator: 5 Order: 4
The smallest generator x the order = 20 (n, Zmodn)
Brittany
1. T/F A field is a set with 2 binary operations that close an Abelian Group.
T
2. Create a Cayley Table showing the pattern of attractivity of (0 bar,2 bar,4 bar) to Zmod6
0 1 2 3 4 5
0 0 1 2 3 4 5
2 0 2 4 0 2 4
4 0 4 2 0 4 2
3. Prove that if Zmodn is a field, iff n is prime. (SBWOC)
Suppose Zmodn is a field.
SBWOC
Suppose n is not prime, that mean n is composite.
Thus for some k,h in Z
h*k=n
Thus, k bar * h bar is in 0 bar
because n bar is in 0 bar
which is not possible in a field
Thus n is prime
Carina
1.
Define pointwise multiplication. Do this by giving a general
case as well as a specific case.
2.
What is the difference between a field and a ring? Create a
table of each to help prove your case.
**Also in the same vein, you could give a couple of tables and
ask which tables are fields, rings, neither and have us prove why.
3.
Prove that ideals are to rings as normal subgroups are to
groups. Prove your thought process by giving a general and a concrete
example.
Diane
1. Q: What is the difference between a ring and a field?
A: A ring is a set with 2 binary operations (usually + and *) which has the following properties: additive associativity, additive commutativity, additive
inverse, additive identity, left & right distributivity, and multiplicative associativity. A field has all of the above properties as well as the following ones:
multiplicative commutativity, multiplicative identity, and multiplicative inverse.
2. Q: Find all the ideals for Zmod15.
A: {0, 3, 6, 9, 12} & {0, 5, 10}
3. Q: Choose one of the ideals of Zmod15 and prove that it is an ideal.
Math 361 Test Question Prep
A: Proof of ideals in class notes for 4-18-7 & 4-20-07.
Zach
1. Define pointwise multiplication in fashion parallel with the definition of pointwise addition.
2. How can you tell if Zmodn will be a field or merely a Ring?
3. (R, +, ∙) = Zmod10, I = {0(bar), 4(bar)}
Prove or disprove I is an ideal in Zmod10.
Brian
Question 1: Construct a table that shows Zmod4xZmod4
Answer 1: 0|1|2|3|
---------0|0|0|0|0|
------------1|0|1|2|3
------------2|0|2|0|2
-------------3|0|3|2|1
Question 2: Is Zmod13 a Field, Ring, or Neither? Why?
Answer 2: Zmod13 is a field because 13 is a prime number, which makes it a field because it has no divisors of 0.
Question 3: List the ideals of Zmod 8, not including trivial ones.
Answer 3: The ideals of Zmod8 are Zmod2 and Zmod4 if you don't include the trivial Zmod1 and Zmod8 which we know work for this number
automatically.
Vicky C
1. Define: rings, fields, and ideals.
2. Prove or disprove I is an ideal in Zmod10.
(R,+,*) = Zmod10, I={0,2,4,6,8} (note: numbers in I have bars over them)
3. Consider the Set:
D2={f(x,y): (x,y) is an element of RxR and f is twice differentiable on R x R}
a. Prove or disprove that D2 is a normal subgroup of F.
b. Give an example of a non-trivial coset involving D2.
c. True or false; D2 is a subgroup of D1.
Vicky B
1. Suppose R  2  2 matrices over . Is R,, a ring, a field, or neither? Explain your answer.
2. Suppose R  f f is a continuous function defined on (-2,2). Create a non-trivial ideal and prove that it is
indeed an ideal.
3. Explain the difference between a ring and a field and how this partitions Z mod n .
Ling Ya
Test questions for Math 361—
1. List the properties of fields and show an example of fields?
Math 361 Test Question Prep
Solution: Properties of fields:
Both + and * are associative
Both + and * are commutative
Multiplication distributes over addition
Zero is the additive identity, and one is the multiplicative identity, and these two constants are
different
Existence of additive and multiplicative inverses
An example of fields: the real numbers R, under the usual operations of addition and multiplication
2. Tell true or false and state the reason.
a. Z mod 22 is a field.
b. Complex numbers C with +, * are fields.
c. Polynomials R(x, y) are both rings and fields.
d. Square n×n matrices are rings.
e. The set of all continuous real-valued functions defined on the interval [a, b] is both a ring and a field.
Solution: a. F. Z mod 22 lacks multiplicative inverse, so it is a ring but not a field.
b. T.
c. F. R(x, y) lacks multiplicative inverse and identity, thus R is a ring but not a field.
d. T.
e. F. The set lacks multiplicative inverse and identity, thus it is a ring but not a field.
3. List the properties that fields have to satisfy but not rings?
Solution: Multiplicative commutativity
Multiplicative identity
Multiplicative inverse
Josh
1) If R is a ring, and I is a subset of that ring, then what properties must I have to be an ideal?
i) I  R
ii) I is a normal subgroup of R
iii) Every element in R and I pick one for each and multiple and it will exist in I
2) Prove that Zmod8 is not a field.
Math 361 Test Question Prep
There is no inverse in by which 2  a1  1 for any a in Zmod8.
3) Show that the integers are a ring.

i) (Z, +) is an abelian group defended under addition on the integers
(a+b)+c = a+(b+c) where a,b,c  Z
ii) (Z, ) is a monoid
(a  b)  c  a  (b  c) where a,b,c  Z
iii) Multiplication distributes over addition
Where a,b,c  Z
a·(b + c) = (a·b) + (a·c)

(a + b)·c = (a·c) + (b·c)
Kristen
1.Find and ideal for  mod 12 and prove it is actually and ideal.


Solution:  mod 12  0,1,2,3,4,5,6,7,8,9,10,11


I = 0,2,4,6,8,10
Proof. 1) I  R because x  I , x  R
2) Normal Subgroup
i) closure x, y  I , x  y  I
ii) inverse x, y  I , x  y  0
iii) Anytime x  I and r  R, xr  I
(any even multiplied by an odd is even and thus back in the ideal)
2. Create a Cayley table for Zmod 8 and explain if it is a ring or a field and why. Then use the table to solve the
following if possible x 2  0 , 4x  0 .
Solution: (  8 ,,)

0
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
0
1
2
3
4
5
6
7
0
1
0
2
4
6
0
3
6
1
4
7
2
5
0
4
0
5
2
7
4
1
6
3
0
6
4
2
0
7
6
5
4
3
2
2
3
4
5
6
7
0
2
4
6
0
4
0
4
0
4
0
6
4
2
1
Zmod 8 is a ring but not a field because it is an Abelian group under + and there are missing inverses under  ie
2 ,4 ,6 .
3. Give an example of a field and prove it is so.
Math 361 Test Question Prep
Solution: ( 3 ,,)

0
0
0
0
0
1
2
1
2
0
1
0
2
1
2
Proof. 1)  3 is a ring because it is an Abelian group under +
2)  3 is a field because x, y  Z 3 , x  y  1
Breann
1. Define what a ring is, and how you can tell if a given set is a ring or a field.
2. Show whether or not the following groups are rings or fields:
Zmod5, Zmod6
3. Define ideals and find the ideals for the following rings: Zmod 20, Zmod6.
Nick
1. What is the definition of an Ideal?
-Definition: Given a commutative ring R, a subring
multiplication; that is, if
is said to be an ideal of R if it absorbs
.
To verify that a subset of a commutative ring is an ideal, it is only necessary to check that it is closed under
subtraction and that it absorbs multiplication.
2. Draw a diagram comparing the relationship between Normal Subgroups to Ideals, and comparing it to
Groups and Rings. Explain the relationship.
-Draw a large circle labeled as Rings, and within it draw a small circle and label it Ideals. Now Draw a large
circle and label it Groups, and label the a small circle inside it as Normal Subgroups. Meaning normal
subgroups are to ideals as groups are to rings.3. Is Zmod(10) an Ideal?
Yes.
Adalaide
1: Explain what and ideal is.
Math 361 Test Question Prep
I of a ring (R, +, ∙)
i)
I is a subset of R
ii)
I must be a normal subgroup of (R, +)
iii)
Any time xI and r R , xr I.
2: Write out all the ideas for Zmod12.
{0} → order 1
{0,6} → order 2
{0,4,8} → order 3
{0,3,6,9} → order 4
{0,2,4,6,8} → order 5
3: Prove: Theorem: If k divides n, then there exists an ideal of Zmodn of order k.
Proof. Assume k|n and for arbitrary n, Zmodn is a ring. Suppose k|n. There exists mℤ such that km = n. Use m
(with bar over it) to generate an ideal: (with k elements in it).
I = {0m, 1m, 2m, …,(k-1)m}
= {0, m, 2m, …, (k-1)m}
i)
I is a subset of Zmodn
ii)
I is a normal subgroup of Zmodn, +.
a. Normal is a freebie since Zmodn is Abelian.
b. Subgroup:
i. Closure
ii. Inverse
Brendan
(R,+,∙) = Zmod5
I = {0,3}
T/F: I is an ideal in Zmod5.
1)
Solution: False. I lacks closure
(R,+,∙) = Zmod6
I = {0,2}
T/F: I is an ideal in Zmod6
2)
Solution: False. Again, I lacks closure
(R,+,∙) = Zmod5
I=?
Is there a possible ideal in Zmod5?
3)
Solution: No. For all Zmodn, if n is prime, then there does not exist an ideal for the modulo.
Marisa
1. How do you show that a nonempty subset I of a ring R is a (two-sided) ideal of R?
a. I has to be a subset of R.
b. I has to be a subgroup of R under commutative addition.
c. For all r in R, and i in I, ri is in I and ir is in I.
Math 361 Test Question Prep
2. Write out the ideals for the group Zmod10.
3. Compare/contrast ideals with normal subgroups.