Commutative Ring Theory/Divisibility and principal ideals

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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 .

Definition (similarity):

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.