An Example of a Ring with Zero Divisors
A classical example of a ring with zero divisors is the ring Z10, that is, the ring of integers modulo 10.
The ring Z10 is equipped with two binary operations:
- addition modulo 10 (+)
- multiplication modulo 10 (⋅)
Example
| $$ 4 + 5 = 9 \\ 1 + 9 = 0 \\ 2 + 9 = 1 \\ 3 + 9 = 2 \\ 4 + 9 = 3 $$ | $$ 4 \cdot 5 = 20 \equiv 0 \pmod{10} \\ 1 \cdot 9 = 9 \equiv 9 \pmod{10} \\ 2 \cdot 9 = 18 \equiv 8 \pmod{10} \\ 3 \cdot 9 = 27 \equiv 7 \pmod{10} \\ 4 \cdot 9 = 36 \equiv 6 \pmod{10} $$ |
Note. In modular arithmetic modulo b, the sum of two integers is defined as the remainder obtained after division by b. For example, when working modulo 10, we have 8 + 1 = 9, while 9 + 1 = 0, since 10 is a multiple of 10 and therefore leaves remainder 0.
The structure Z10 is a ring, since it satisfies the same algebraic axioms as the integers with respect to addition and multiplication.
However, the ring Z10 is not an integral domain, because it contains zero divisors.
Indeed, there exist nonzero elements in Z10 whose product is congruent to zero modulo 10.
$$ 2 \cdot 5 = 10 \equiv 0 \pmod{10} $$
$$ 4 \cdot 5 = 20 \equiv 0 \pmod{10} $$
$$ 6 \cdot 5 = 30 \equiv 0 \pmod{10} $$
$$ 8 \cdot 5 = 40 \equiv 0 \pmod{10} $$
Therefore, the elements 2, 4, 5, 6, and 8 are zero divisors in the ring Z10.
More generally, any ring that contains nonzero elements whose product is zero fails to be an integral domain.
