Elementary Abstract Algebra- Examples and Applications, 2019a

3.4 THE INTEGERS MOD N (A.K.A. Z N ) 121
3.4.4 Identities and inverses in Z n
Next, we want to look at some additional properties that were introduced in
Chapter 2, namely identities and inverses (both additive and multiplicative).
This time we’ll go through these properties more quickly.
Consider first the additive identity. Remember that an additive identity
is an element which, when added to any other element a, gives a result of a.
For the specific case of Z 8 , we can see from the first row of Table 3.3 that
0 ⊕ a = a for any a ∈ Z 8 . Similarly, the first column of Table 3.3 show that
a ⊕ 0=a for any a ∈ Z 8 .
Is 0 an additive identity for any Z n ? Not surprisingly, the answer is Yes:
Proposition 3.4.17. 0 ∈ Z n is the additive identity of Z n .
Proof. Given any a ∈ Z n ,thena ⊕ 0 is computed by taking the remainder
of a+0 mod n. Sincea+0 = a, and0≤ a1. What is the multiplicative identity for Z n when n =1?
♦
3.4.5 Inverses in Z n
Now let’s find out whether the integers mod n have additive and multiplicative
inverses. Additive inverse first: for each element of Z n is there a
corresponding element of Z 8 such that their modular sum is the additive
identity (that is, 0)? You may see in Table 3.3 that each row of the addition
table contains a 0 (e.g. 1 ⊕ 7 = 0). It follows that each element of Z 8 has an
additive inverse. But will the same be true for Z 27 ,orZ 341 ,orZ 5280 ? We
can’t just take this for granted–we need to give a proof:
Proposition 3.4.19. Let Z n be the integers mod n and a ∈ Z n .Thenfor
every a there is an additive inverse a ′ ∈ Z n .
In other words: for any a ∈ Z n in we can find an a ′ such that:
a ⊕ a ′ = a ′ ⊕ a =0.