# Commutative Ring Theory/Divisibility and principal ideals

**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.