Associate elements

Definition

Two elements in a commutative unital ring are said to be associate elements if they satisfy the following:

• Each element is a divisor of the other element.
• The principal ideal generated by one element equals the principal ideal generated by the other.

The relation of being associate elements is an equivalence relation.

For full proof, refer: Associate element relation is an equivalence relation

Facts

In an integral domain, two elements are associate if and only if they are in the same orbit under the multiplicative action of the group of units of the ring (or in other words, there is an invertible element that multiplied with the first gives the second). Refer:

• Elements in same orbit under multiplication by group of units are associate
• Associate implies same orbit under multiplication by group of units in integral domain
• Associate not implies same orbit under multiplication by group of units