MAE 301
Notes for September 1, 2010
By Jillian Stark
Properties of a Group, Ring, and
Field
Definition-Group: a set with a binary operation denoted by *, such that…..
4 Properties a group must satisfy:
1. The set must be closed. If a,b EG then a*bG, therefore the set is closed
2.  an identity element, e, such that for all gG, g*e=g=e*g
3. The operation * is associative such that if a,b,cEG, (a*b)*c=a*(b*c) so the
operation * is associative on G
4. For each gG  an inverse g-1 in G such that g-1*g=g-1*g=e
Note: A group does not have to be commutative. However, if it is a commutative
group, it then becomes known as an abelian group.
The set is commutative if for all a,bEG , a+b=b+a.
Definition-Abelian Group: A group is an abelian group if and only if the group
operation is commutative
Examples:
Some examples of Groups:
(Let us assume the following include zero)
i. (N, +) Is this set closed under addition?-Yes
 identity element?-Yes
Associative?-Yes
Are there inverses?-No (This is because there are not negative
numbers and therefore there are no additive inverses, i.e. 2+ ?=0 ?=-2 but -2 is
not in the set of natural numbers, N)
Therefore this is not a group because it does not satisfy all 4 Properties of a group
ii.
(Z, +) Is this set closed under addition?-Yes
 identity element?-Yes (0 is the identity element because when
you add 0 to any integer it preserves that particular integer, i.e. let a=an
integer, then a+0=0+a=a
Associative?-Yes
Does each element have an inverse?-Yes(Includes negatives)
Therefore this is a group because it satisfies all 4 Properties of a group
*Note: This is an abelian group because this set is commutative*
iii.
(Z, ) Is this set closed under multiplication?-Yes
 identity element?-Yes(1 is the identity element because and
integer multiplied by 0 equals 0, I,e, a*1=1*a=a)
Associative?-Yes
Are there inverses?-No (No multiplicative inverse for zero)
Definition-Ring: a Ring, R, is a set closed under 2 binary operations (+,)
The following axioms of a ring must be satisfied:
1. (R,+) is an abelian group (assume commutativity). For all a,b,c  R:
a. there is an identity element 0  R such that a+0=0+a=a1
b. for every a  R there is an additive inverse –a  R such that
a + (-a)=(-a) +a=0
c. a + (b+c)=(a+b) +c
d. a +b=b+ a
2. The operation ,, is associative(do not assume commutativity). For all a,b,c 
R,
a (b  c)=(a  b) c
3. The multiplication operation is distributive over addition. For all a,b,c  R
a (b+c)= (a b) + (a c) and (a+b) c=(a  c) + (b  c) (1.Smith)
-There sometimes exists a multiplicative identity 1, such that for every rR,
1*r=r*1=r
Examples :
Examples of a Ring:
i. (Reals, +,  )
ii. (Z, +,  )
iii. (Q, +,  )
iv. (N U 0 , +, )-NOT a ring because (N U 0 , +, ) is not a group ( only 0 has
an additive inverse
Definition- Commutative Ring: (R, +, ) is a commutative ring if and only if for all
a,b  R, ab=ba.
Definition- Field: A commutative ring with multiplicative identity of 1, in which all
non-zero elements are invertible.
Examples:
Examples of a Field:
i. The ring of rational numbers and the ring of real numbers are fields
Transition to Advance Mathematics. Douglas Smith. Cengage Learning. Pages 306311
1
ii. The ring of Z is NOT a field because no element (other than 1 and -1)has a
multiplicative inverse in Z.
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
Complex Numbers (C)
C=a+bi, a,b  Reals
Review of i:
i= 1
i2=-1
i3=-i
i4=1
Question: Are all complex numbers invertible?
Problem:Find the inverse of 5+i
Let us find the multiplicative inverse of (5+i) such that (5+i)*(inverse)=1
Allow c,d to be some rational numbers.
(5+i)(c+di)=1 When we multiply the left side out we get:
5c+5di+ci-d=1 Let us now combine the imaginary numbers 5di and ci:
5c-d +(5d+c)i=1 We know that the real number in this equation (5c-d) must equal 1
and we know that the imaginary number 5d+c must equal zero because we want the
imaginary part to go away so that we are left with 1:
5c-d=1 5d+c=0 Solve for c and d:
5
1
c=
d=
26
26
5
1
 i
inverse of 5+i is
26 26
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Definition(Suggested in class)- Prime Number: A positive integer with exactly 2
distinct positive factors or such a number’s additive inverse

Examples:
Examples of prime numbers include: 3, 5, 7, 11…….
2 Different Notions in High School:
1. Prime->relates to divisibility
2. Irreducible->relates to factorization
Definition- Algebraic Numbers: Set of solutions to non-zero polynomial equations
with rational coefficients
Examples:
i. x2-2=0 x=+/- 2 = algebraic number because it satisfies the non-zero polynomial
equation to its left.

Definition- Transcendental Numbers: All complex numbers that are not algebraic
Definition- Equivalence Relation “”: An equivalence relation on a set, S, is one that
satisfies:
1. Reflexivity: sS, ss
2. Symmetry: If st then ts
3. Transitivity: s,t,v  S (st and tv) sv
Examples:
Examples of Equivalence Relations:
i. Z(integers),  : nn-Yes it is reflexive because n=n, if nm, mn-No( because
if n is greater than m, then m cannot be greater than n. i.e. 32 then 23
cannot hold), nm, mpnp-Yes, for example let 32 and 21 then 31
holds
Not all 3 properties are satisfied so this is NOT an equivalence relation. Thus
the values on each side of the sign  cannot be substituted for one another.
ii. Similarity is an equivalence relation on the set of plane figures. One object is
congruent to the scaling (enlarging or shrinking) of another. (Angles and
corresponding sides for example). Examples of two triangles being
similar include: AA (Angle Angle) is when two triangles have two angles
that are corresponding that also have the same measurements(obviously
the third angle for both triangles is the same as well). SSS (Side Side Side)
is when two triangles have the same ratio of corresponding sides no
matter which sides, then the triangles are identical in shape.
Question for the Class:
What is the importance of a unit? What are the units in various sets?
In class the term “unit” was defined. I was unsure as to its importance with the
lesson, but while researching prime numbers I found that it was quite important.
Definition- Unit: An invertible element is called a unit, also known as a divisor of
unity. These elements have multiplicative inverses. Within the set of integers there
are usually 2 units 1 and -1. If we were to include Gaussian integers a+bi, a,bZ
then there are 4 units 1, -1, i, and –i. In other sets there can be indefinitely many
units. (2. Primes.utm.edu)
Also after the lecture I wanted to find out why the number one is not considered a
prime number. When I was researching I found that units cannot be primes. I also
found it interesting that there are infinitely many prime numbers. To prove this you
could use a proof by contradiction.
There are infinitely many primes
Proof:
Assume NOT
Assume there are a finite number of primes, a1, a2, a3, ………..an
Now let there be some number (a1 *a2* a3,*…..an) + 1.
Every prime number when divided into this number will have a remainder of 1.
Therefore this number has 0 prime factors which is a CONTRADICTION
There are infinitely many prime numbers
End of Proof.
Problem concerning commutative rings (fields)
Does the set of 3x3 matrices of the form below form a commutative ring?
r 0 0
0 r 0
s t r
Hint: Check if it is closed and commutative under multiplication

r 0 0
0 r 0
s
t
a 0 0
0 a 0=
r
b c
ra
0
0
ra
0
0
sa  rb ta  rc
a
ra
this does show that the set is closed under multiplication



a 0 0 r 0 0
0 a 0  0 r 0=
b c

a
s
t


Therefore ………….
ar
0
0
ar
0
0
r br  as cr  at ar
r 0 0
0 r 0
s
t
r
a 0 0 a 0 0 r 0 0
 0 a 0 = 0 a 0 0 r 0
b c
a
b c
a
Answer: Yes, it is commutative! It is a commutative ring.




s
t
r (commutativity)