Definition (principal ideal):
Let be a commutative ring. A principal ideal is a left principal ideal of . Equivalently, it is a right principal ideal or a two-sided principal ideal of .
Proposition (characterisation of divisibility by principal ideals):
Let be a commutative ring, and let . Then .
Proof: Both assertions are equivalent to the existence of a such that .
Let be a commutative ring. Two elements are called similar if and only if there exists a unit such that .
Proposition (similarity is an equivalence relation):
Given a ring , the relation of similarity defines an equivalence relation on the elements of .
Proof: For reflexivity, use the identity, and for symmetry, use the inverse. Suppose that and , where . Then , where of course .
Proposition (in an integral domain, the generating element of a principal ideal is unique up to similarity):
Let be an integral domain, and let be a principal ideal of . Then if for some element , we have for some .
Proof: The equation implies that and for certain . Hence, . By cancellation (which is applicable because is an integral domain), and hence is a unit, so that and are similar.