MATH 57091 - Algebra for High School Teachers
Algebraic Properties of Zm
Professor Donald L. White
Department of Mathematical Sciences
Kent State University
D.L. White (Kent State University)
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Examples
+
0
1
2
3
0
0
1
2
3
Z4
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
×
0
1
2
3
0
0
0
0
0
Z4
1
0
1
2
3
2
0
2
0
2
3
0
3
2
1
Observe the following:
Each table is symmetric with respect to the diagonal.
Conclusion: Addition and multiplication are commutative.
Each table has a row equal to the row of labels.
Conclusion: There is a neutral element or identity element
for addition (0) and for multiplication (1).
There is a 0 in every row of the addition table.
Conclusion: Each element has an additive inverse.
Not every row of the multiplication table contains a 1.
Conclusion: Not every element of Z4 has a multiplicative inverse.
D.L. White (Kent State University)
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Examples
Contrast the multiplication tables of Z4 and Z5 :
×
0
1
2
3
0
0
0
0
0
Z4
1
0
1
2
3
2
0
2
0
2
3
0
3
2
1
×
0
1
2
3
4
0
0
0
0
0
0
Z5
1 2
0 0
1 2
2 4
3 1
4 3
3
0
3
1
4
2
4
0
4
3
2
1
Every non-zero row of the multiplication table for Z5 does contain a 1.
Conclusion: Every non-zero element of Z5 has a multiplicative inverse.
There is a 0 in the table for Z4 , in neither the 0 row nor the 0 column.
Conclusion: Products of non-zero elements can be 0!
This last observation means we will need to re-think some procedures,
such as solving polynomial equations by factoring.
D.L. White (Kent State University)
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Commutativity, Associativity, Distributivity
We now state some of our observations formally.
Theorem
Let m be a fixed positive integer.
For all [a], [b], [c], and [d] in Zm , the following hold:
i
[Commutativity of Addition] [a] + [b] = [b] + [a]
ii
[Commutativity of Multiplication] [a] · [b] = [b] · [a]
iii
[Associativity of Addition] [a] + ([b] + [c]) = ([a] + [b]) + [c]
iv
[Associativity of Multiplication] [a] · ([b] · [c]) = ([a] · [b]) · [c]
v
[Distributivity] [a] · ([b] + [c]) = [a] · [b] + [a] · [c]
All of these properties of Zm follow from the definitions of the operations
and the same properties of the set Z of integers.
We will prove some and leave others as exercises.
D.L. White (Kent State University)
4/7
Commutativity, Associativity, Distributivity
Proof: (i) and (iv) are homework problems; (ii) is proved in the text.
iii Associativity of Addition:
[a] + ([b] + [c]) = [a] + [b + c] by addition in Zm ,
= [a + (b + c)] by addition in Zm ,
= [(a + b) + c] by associativity of addition in Z,
= [a + b] + [c] by addition in Zm ,
= ([a] + [b]) + [c] by addition in Zm .
v
Distributivity:
[a] · ([b] + [c]) = [a] · [b + c] by addition in Zm ,
= [a(b + c)] by multiplication in Zm ,
= [ab + ac] by distributivity in Z,
= [ab] + [ac] by addition in Zm ,
= [a] · [b] + [a] · [c] by multiplication in Zm . D.L. White (Kent State University)
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Identities and Additive Inverses
Theorem
Let m be a fixed positive integer.
For all [a] in Zm , the following hold:
i
[0] + [a] = [a];
[0] is an identity element for addition, or additive identity.
ii
[1] · [a] = [a];
[1] is an identity element for multiplication, or multiplicative identity.
Both properties follow immediately from the definitions of the operations
and the analogous properties of 0 and 1 in Z.
Theorem
Let m be a fixed positive integer.
For all [a] in Zm , [a] + [−a] = [0]; that is, [−a] is an additive inverse for [a].
This also follows from the definition and the fact that a + (−a) = 0 in Z.
D.L. White (Kent State University)
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Identities and Additive Inverses
NOTES:
Since m − a ≡ −a (mod m), we have [m − a] = [−a],
and so the additive inverse of [a] in Zm can be written as [m − a].
For example, in Z15 we can write the additive inverse of [9]
as [−9] or [15 − 9] = [6], whichever is more convenient.
We have [9] + [6] = [15] = [0] since 15 ≡ 0 (mod 15).
We usually denote the additive inverse of [a] by −[a],
and so −[a] = [−a] = [m − a].
We can then define subtraction in Zm by [a] − [b] = [a] + (−[b]).
The concept of “positive” or “negative” does not make sense in Zm .
For example, in Z15 , [9] = [−6] and −[9] = [−9] = [6].
The properties in the theorems above imply that Zm is a ring.
We will discuss the general concept of a ring later.
Not all elements of Zm have multiplicative inverses in general.
Try to determine which elements of Zm have multiplicative inverses.
D.L. White (Kent State University)
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