# Commutative Ring Theory/Printable version

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# 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.

# Greatest common divisors

**Definition (divisor)**:

Let be a ring, and let . A **divisor** of is an element such that there exists such that . The notation indicates that is a divisor of

**Definition (greatest common divisor)**:

Let be a commutative ring, and let . A **greatest common divisor** is an element such that for all , and such that for any other element such that for all , we have .

**Definition (coprime)**:

Let be a commutative ring, and let . These elements are said to be **coprime** if and only if whenever is such that for all , then .

**Proposition (a set of elements of a commutative ring divided by their greatest common divisor is coprime)**:

Let

# Bézout domains

**Definition (Bézout domain)**:

A **Bézout domain** is an integral domain whose every finitely generated ideal is principal, ie. generated by a single element.

**Proposition (Every Bézout domain is a GCD domain)**:

Let be a Bézout domain. Then is a GCD domain.

**Proof:** Given any two elements , we may consider the ideal generated by and . By the definition of Bézout domains, for at least one (which is moreover unique up to similarity). Then and , so that by the characterisation of divisibility by principal ideals, is a common divisor of and . Moreover, if is another common divisor of and , then , so that , so that . Hence, is a greatest common divisor of and .

# Principal ideal domains

**Definition (principal ideal domain)**:

A **principal ideal domain** is an integral domain whose every ideal is principal.

**Proposition (a Bézout domain is principal if and only if it is Noetherian or satisfies the ascending chain condition for principal ideals)**:

Let be a Bézout domain. Then the following are equivalent:

- is a principal ideal domain
- is noetherian
- The principal ideals of satisfy the ascending chain condition

**Proof:** The implication "1. 2." is obvious. Suppose that 3. holds, and let be any ideal. If was non-principal, then whenever , we could find a such that . Hence, starting with an arbitrary and invoking the axiom of dependent choice (applied to a set of finite tuples with an adequate relation) yields a sequence in such that ; indeed, since is a Bézout domain. If we define

- , we have ;

thus, we have defined an ascending chain of principal ideals of that does not stabilize. Finally, every principal ideal domain must be noetherian, since being noetherian is equivalent to all ideals being finitely generated.

# Derivations

**Proposition (alternative construction of the universal derivation)**:

Let be a unital -algebra. Note that becomes an -module via the linear extension of the operation . We then have a morphism of -modules

- ,

where the dot indicates the algebra multiplication of . Set and . Then

is a derivation, and we have an isomorphism inducing a commutative diagram

**Proof:** Note first that is a derivation. This takes some explaining. First, note that for arbitrary the element is in . Moreover, from this follows that the element

is in for arbitrary.

Hence, from the universal property of , we obtain a unique morphism of -modules that makes the diagram

commutative. We construct an inverse map to . Namely, on we can define the map