Let n be an integer. Consider Z/nZ, the set of equivalence classes of Z modulo n. Addition and multiplication with these classes can be defined in the following way.
On Z/nZ we define two so-called binary operations, an addition and a multiplication, by:
a (mod n) · b (mod n) = (a · b) (mod n).
(We usually omit the multiplication dot.)
A neutral element for the addition is 0 (mod n). Indeed,
A neutral element for the multiplication is 1 (mod n) as we have
a (mod n) · 1 (mod n) = a (mod n) .
The set Z/nZ together with addition and multiplication is an example of a quotient ring, a structure to be discussed in Chapter 7.