Commutative Ring Theory/Printable version

Commutative Ring Theory

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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):


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:

  1.   is a principal ideal domain
  2.   is noetherian
  3. The principal ideals of   satisfy the ascending chain condition
(On the condition of the axiom of dependent choice.)

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.  


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