Commutative Ring Theory/Divisibility and principal ideals

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 . characterisation of divisibility by principal ideals Let be a commutative ring, and let . Then . Both assertions are equivalent to the existence of a such that . similarity Let be a commutative ring. Two elements are called similar if and only if there exists a unit such that . similarity is an equivalence relation Given a ring , the relation of similarity defines an equivalence relation on the elements of . For reflexivity, use the identity, and for symmetry, use the inverse. Suppose that and , where . Then , where of course . 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 . 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.