ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

What is said above may be generalized to polynomials, whos coefficients are
taken from an arbitrary ring. This may lead to finite fields.
Example 58: Rings of remainder classes are the prototypes of a quotient ring.
(see nr 129)
Let n ∈ N with n > 1.
Let I = nZ, and notice that I is an ideal in the ring (Z, +, ·). Two elements
x and y are equivalent if y ∈ x + I, that is when y − x ∈ nZ. Let k be the
canonical projection of Z on Z/I, defined by k(x) = [x] = x + I = x + nZ,
which means that k(x) is the remainder class modulo n which contains x.
We shall here see an alternative way to construct this ring:
Let Zn = {0, . . . , n − 1}. We define an addition on Zn by letting u ⊕ v =
(u + v) mod n. Analogously we define subtraction and multiplication. There
are also obvious candidates for 0-element and 1-element. We are going to show
that this defines a ring isomorphic with Z/nZ.
It is seen that the mapping f : Z → Zn, defined by f(z) = z mod n, is a
homomorphism with respect to the underlying algebraic structure. To check
the axioms we us that f is surjective to see that addition and multiplication
are associative. The remaining axioms are straightforward.
The mapping g : Zn → Z/I defined by g(x) = [x], is en bijection. Its inverse
h = g −1 is given by h([x]) = x mod n.
Direct check gives that f = h ◦ k. Then f and k are homomorphisms and since
f is surjective it follows that h is a homomorphism and consequently an isomorphism.
OBS: indsÆt om muligt reference !!!It may be noticed that instead
of using the classes we are using the principal representatives and in stead of
calculating with classes we are doing calculation with the representatives. You
can see an example of multiplication tables on the figure: In praxis you wont
look in the table for each operation but instead carry out the operations in the
usual way and then take the remainder at last or at certain practical moments.
0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
81
0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1