Zero Divisors
In a commutative ring (S,+,·), a nonzero element a≠0 in the set S is called a zero divisor if there exists another nonzero element b≠0 in S such that ab = 0 $$ \exists \ \ a,b \in S \ , \ a \ne 0 \ , \ b \ne 0 \ \ \ | \ \ \ a \cdot b = 0 $$
The commutative ring of real numbers (R,+,*) contains no zero divisors.
That’s why the concept can feel counterintuitive at first.
Note. We’re taught early on - often as soon as middle school - that division by zero is undefined. So it’s not surprising that the idea of zero divisors in abstract algebra tends to cause confusion.
To clarify the concept, let’s explore a concrete example from modular arithmetic.
A Practical Example
Consider the set of equivalence classes modulo 6:
$$ Z_6 = \{ 0,1,2,3,4,5 \} $$
This set, Z₆, forms a commutative ring (Z₆,+,·) under addition and multiplication:
$$ (Z_6,+, \cdot) $$
To determine whether zero divisors exist in this structure, we can construct its multiplication table:
| a ·6 b | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 0 | 2 | 4 | 0 | 2 | 4 |
| 3 | 0 | 3 | 0 | 3 | 0 | 3 |
| 4 | 0 | 4 | 2 |
0 | 4 | 2 |
| 5 | 0 | 5 | 4 | 3 | 2 | 1 |
In this ring, the elements 2, 3, and 4 are zero divisors.
Take 2 and 3, for instance:
$$ 2 \cdot 3 \equiv 0 \mod 6 $$
Explanation. First, compute the product: $$ 2 \cdot 3 = 6 $$ Then reduce modulo 6: $$ 6 \div 6 = 1 \ \ r = 0 $$ Since the remainder is zero, we have 2·3 ≡ 0 mod 6. The same holds for 3·2.
Likewise, 4 is a zero divisor because
$$ 4 \cdot 3 \equiv 0 \mod 6 $$
Explanation. Multiply 4 and 3: $$ 4 \cdot 3 = 12 $$ Now reduce modulo 6: $$ 12 \div 6 = 2 \ \ r = 0 $$ The remainder is zero again, so 4·3 ≡ 0 mod 6 - and, again, the same goes for 3·4.
Example 2
Now let’s turn to the ring of real numbers (R,+,·):
$$ (R,+,*) $$
In this ring, the product of any two nonzero real numbers a≠0, b≠0 is never zero:
$$ \forall \ a,b \in R \ , \ a \ne 0 \ , \ b \ne 0 \ \ \Longrightarrow \ ab \ne 0 $$
That is, if the product ab equals zero, then at least one of the factors must be zero:
$$ ab=0 \ \ \Longrightarrow \ a=0 \ \text{or} \ b = 0 $$
Therefore, the ring of real numbers (R,+,·) contains no zero divisors.
And so on.
