Associates

Let a and b be two elements of the set of integers Z. The two integers are said to be associates if each is a divisor of the other. $$ a|b \\ b|a $$

Two nonzero integers are associates if they differ only by a sign.

Every nonzero integer is an associate of its additive inverse.

For example, 4 and -4, or 8 and -8, and so on.

The associates form equivalence classes, each consisting of exactly two nonzero integers.

Note. Only the equivalence class of zero consists of a single element, because division by zero is not defined in the ring of integers.

Proof

The integer a is a divisor of b if there exists an integer k such that

$$ ak = b $$

Similarly, the integer b is a divisor of a if there exists an integer j such that

$$ bj = a $$

It follows that the integers k and j must necessarily be units of Z.

$$ ak = bj $$

$$ a \cdot u = b \cdot u $$

where u can take only the values 1 or -1.

Therefore, every nonzero integer is an associate of its additive inverse.

A concrete example

Consider the integer a = 4 and the integer b = 8.

We can state that a is a divisor of b

$$ a|b = 4|8 $$

However, we cannot state that b is a divisor of a.

We must therefore look for integers that divide each other.

Let us try a = 4 and b = 4.

In this case, we can state that

$$ a|b = 4|4 $$

Note. There exists an integer k = 1 such that ak = b. $$ a \cdot 1 = b $$

$$ b|a = 4|4 $$

Note. There exists an integer j = 1 such that bj = a. $$ b \cdot 1 = a $$

Therefore, every integer is an associate of itself.

This, however, is a trivial solution. We are interested in associates that are distinct integers.

It is nevertheless straightforward to see that a = 4 and b = -4 are also associates.

They are distinct integers, a ≠ b, and each divides the other.

$$ a|b = 4|-4 $$

Note. There exists an integer k = -1 such that ak = b. $$ a \cdot -1 = b $$

$$ b|a = -4|4 $$

Note. There exists an integer j = -1 such that bj = a. $$ b \cdot -1 = a $$

Thus, every integer is an associate of its additive inverse.

And so on.