Commutative Algebra/Jacobson rings and Jacobson spaces

Definition and elementary characterisations

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Definition 14.1:

A Jacobson ring is a ring such that every prime ideal is the intersection of some maximal ideals.

Before we strive for a characterisation of Jacobson rings, we shall prove a lemma first which will be of great use in one of the proofs in that characterisation.

Lemma 14.2:

Let   be a Jacobson ring and let   be an ideal. Then   is a Jacobson ring.

Proof:

Let   be prime. Then   is prime. Hence, according to the hypothesis, we may write

 ,

where the   are all maximal. As   is surjective, we have  . Hence, we have

 ,

where the latter equality follows from   implying that for all  ,  , where   and   and thus  . Since the ideals   are maximal, the claim follows. 

Theorem 14.3:

Let   be a ring. The following are equivalent:

  1.   is a Jacobson ring.
  2. Every radical ideal (see def. 13.1) is an intersection of maximal ideals.
  3. For every   prime the Jacobson radical of   equals the zero ideal.
  4. For every ideal  , the Jacobson radical of   is equal to the nilradical of  .

Proof 1: We prove 1.   2.   3.   4.   1.

1.   2.: Let   be a radical ideal. Due to theorem 13.3,

 .

Now we may write each prime ideal   containing   as the intersection of maximal ideals (we are in a Jacobson ring) and hence obtain 1.   2.

2.   3.: Let   be prime. In particular,   is radical. Hence, we may write

 ,

where the   are maximal. Now suppose that   is contained within the Jacobson radical of  . According to theorem 13.7,   is a unit within  , where   is arbitrary. We want to prove  . Let thus   be such that  . Then   and thus   with   and  , that is  . Let   be the inverse of  , that is  . This means   for all  , and in particular,  . Hence  , contradiction.

3.   4.: Let  . Assume there exists   and a prime ideal   such that  , but   for all maximal  . Let   be the canonical projection. Since preimages of prime ideals under homomorphism are prime,   is prime.

Let   be a maximal ideal within  . Assume  . Let   be the canonical projection. As in the first proof of theorem 12.2,   is maximal.

We claim that   is maximal. Assume  , that is   for a suitable  . Since  ,  , contradiction. Assume   is strictly contained within  . Let  . Then  . If  , then  , contradiction. Hence   and thus  , that is  .

Furthermore, if  , then  . Now   since  . Hence,  , that is,  , a contradiction to  .

Thus,   is contained within the Jacobson radical of  .

4.   1.: Assume   is prime not the intersection of maximal ideals. Then

 .

Hence, there exists an   such that   for every maximal ideal   of  .

The set   is multiplicatively closed. Thus, theorem 12.3 gives us a prime ideal   such that  .

Let   be a maximal ideal of   that does not contain  . Let   be the canonical projection. We claim that   is a maximal ideal containing  . Indeed, the proof runs as in the first proof of theorem 12.2. Furthermore,   does not contain  , for if it did, then  . Thus we obtained a contradiction, which is why every maximal ideal of   contains  .

Since within  , the Jacobson radical equals the Nilradical,   is also contained within all prime ideals of  , in particular within  . Thus we have obtained a contradiction. 

Proof 2: We prove 1.   4.   3.   2.   1.

1.   4.: Due to lemma 3.10,   is a Jacobson ring. Hence, it follows from the representations of theorem 13.3 and def. 13.6, that Nilradical and Jacobson radical of   are equal.

4.   3.: Since   is a radical ideal (since it is even a prime ideal),   has no nilpotent elements and thus it's nilradical vanishes. Since the Jacobson radical of that ring equals the Nilradical due to the hypothesis, we obtain that the Jacobson radical vanishes as well.

3.   2.: I found no shorter path than to combine 3.   1. with 1.   2.

2.   1.: Every prime ideal is radical. 

Remaining arrows:

1.   3.: Let   be a prime ideal of  . Now suppose that   is contained within the Jacobson radical of  . According to theorem 13.7,   is a unit within  , where   is arbitrary. Write

 ,

where the   are maximal. We want to prove  . Let thus   be such that  . Then   and thus   with   and  , that is  . Let   be the inverse of  , that is  . This means   for all  , and in particular,  . Hence  , contradiction.

3.   1.: Let   be prime. If   is maximal, there is nothing to show. If   is not maximal,   is not a field. In this case, there exists a non-unit within  , and hence, by theorem 12.1 or 12.2 (applied to   where   is a non-unit),   contains at least one maximal ideal. Furthermore, the Jacobson radical of   is trivial, which is why there are some maximal ideals   of   such that

 .

As in the first proof of theorem 12.2,   are maximal ideals of  . Furthermore,

 .

2.   4.: Let   be the nilradical of  . We claim that

 .

Let first  , that is,  . Then  , that is   and  . The other inclusion follows similarly, only the order is in reverse (in fact, we just did equivalences).

Due to the assumption, we may write

 ,

where the   are maximal ideals of  .

Since   is surjective,  . Hence,

 ,

where the last equality follows from   implying that   for   and   and hence   for all  . Furthermore, the   are either maximal or equal to  , since any ideal   of   properly containing   contains one element   not contained within  , which is why  , hence   and thus  .

Thus,   is the intersection of some maximal ideals of  , and thus the Jacobson radical of   is contained within it. Since the other inclusion holds in general, we are done.

4.   2.: As before, we have

 .

Let now   be the Jacobson radical of  , that is,

 ,

where the   are the maximal ideals of  . Then we have by the assumption:

 .

Furthermore, as in the first proof of theorem 12.2,   are maximal.

Goldman's criteria

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Now we shall prove two more characterisations of being a Jacobson ring. These were established by Oscar Goldman.

Theorem 14.4 (Goldman's first criterion):

Let   be a ring.   is Jacobson if and only if   is.

This is the hard one, and we do it right away so that we have it done.

Proof:

One direction ( ) isn't too horrible. Let   be a Jacobson ring, and let   be a prime ideal of  . (We shall denote ideals of   with a small zero as opposed to ideals of   to avoid confusion.)

We now define

 .

This ideal contains exactly the polynomials whose constant term is in  . It is prime since

 

as can be seen by comparing the constant coefficients. Since   is Jacobson, for a given   that is not contained within  , and hence not in  , there exists a maximal ideal   containing  , but not containing  . Set  . We claim that   is maximal. Indeed, we have an isomorphism

 

via

 .

Therefore,   is a field if and only if   is. Hence,   is maximal, and it does not contain  . Since thus every element outside   can be separated from   by a maximal ideal,   is a Jacobson ring.

The other direction   is a bit longer.

We have given   a Jacobson ring and want to prove   Jacobson. Hence, let   be a prime ideal, and we want to show it to be the intersection of maximal ideals.

We first treat the case where   and   is an integral domain.

Assume first that   does contain a nonzero element (i.e. is not equal the zero ideal).

Assume   is contained within all maximal ideals containing  , but not within  . Let   such that   is of lowest degree among all nonzero polynomials in  . Since  ,  . Since   is an integral domain, we can form the quotient field  . Then  .

Assume that   is not irreducible in  . Then  ,  , where  ,   are not associated to  . Let   such that  . Then  . As   is prime, wlog.  . Hence  . Thus,   and   are associated, contradiction.

  is Euclidean with the degree as absolute value. Uniqueness of prime factorisation gives a definition of the greatest common divisor. Since   is irreducible in   and  ,  . Applying the Euclidean algorithm,  ,  . Multiplication by an appropriate constant   yields  ,  . Thus,  . Hence,   is contained within every maximal ideal containing  . Further,  .

Let   be any maximal ideal of   not containing  . Set

 .

Assume  . Then  ,  . We divide   by   by applying a polynomial long division algorithm working for elements of a general polynomial ring: We successively eliminate the first coefficient of   by subtracting an appropriate multiple of  . Should that not be possible, we multiply   by the leading coefficient of  , that shall be denoted by  . Then we cannot eliminate the desired coefficient of  , but we can eliminate the desired coefficient of  . Repeating this process gives us

 ,  

for  . Furthermore, since this equation implies  , we must have   since the degree of   was minimal among polynomials in  . Then

 

with  . By moving such coefficients to  , we may assume that no coefficient of   is in  . Further,   is nonzero since otherwise  . Denote the highest coefficient of   by  , and the highest coefficient of   by  . Since the highest coefficients of   and   must cancel out (as  ),

 .

Thus,   and  , but  , which is absurd as every maximal ideal is prime. Hence,  .

According to theorem 12.2, there exists a maximal ideal   containing  . Now   does not equal all of  , since otherwise  . Hence,   and the maximality of   imply  . Further,   is a maximal ideal containing   and thus contains  . Hence,  .

Thus, every maximal ideal   that does not contain   contains  ; that is,   for all maximal ideals   of  . But according to theorem 12.3, we may choose a prime ideal   of   not intersecting the (multiplicatively closed) set  , and since   is a Jacobson ring, there exists a maximal ideal   containing   and not containing  . This is a contradiction.

Let now   be the zero ideal (which is prime within an integral domain). Assume that there are only finitely many elements in   which are irreducible in  , and call them  . The element

 

factors into irreducible elements, but at the same time is not divisible by any of  , since otherwise wlog.

 ,

which is absurd. Thus, there exists at least one further irreducible element not listed in  , and multiplying this by an appropriate constant yields a further element of   irreducible in  .

Let   be irreducible in  . We form the ideal   and define  . We claim that   is prime. Indeed, if  , then   and   factor in   into irreducible components. Since   is a unique factorisation domain,   occurs in at least one of those two factorisations.

Assume there is a nonzero element   contained within all the  , where   is irreducible over  .   factors in   uniquely into finitely many irreducible components, leading to a contradiction to the infinitude of irreducible elements of  . Hence,

 ,

where each   is prime and  . Hence, by the previous case, each   can be written as the intersection of maximal elements, and thus, so can  .

Now for the general case where   is an arbitrary Jacobson ring and   is a general prime ideal of  . Set  .   is a prime ideal, since if  , where  , then   or  , and hence   or  . We further set  . Then we have

 

via the isomorphism

 .

Set

  and  .

Then   is an integral domain and a Jacobson ring (lemma 14.2), and   is a prime ideal of   with the property that  . Hence, by the previous case,

 .

Thus, since  ,

 ,

which is an intersection of maximal ideals due to lemma 12.4 and since isomorphisms preserve maximal ideals. 

Theorem 14.5 (Goldman's second criterion):

A ring   is Jacobson if and only if for every maximal ideal  ,   is maximal in  .

Proof:

The reverse direction   is once again easier.

Let   be a prime ideal within  , and let  . Set

 .

Assume  . Then there exist  ,   such that

 .

By shifting parts of   to  , one may assume that   does not have any coefficients contained within  . Furthermore, if   follows  . Further,  , since if  ,  ,  , then   annihilates all higher coefficients of  , which is why   equals the constant term of   times   and thus  . Hence   and let   be the leading coefficient of  . Since the nontrivial coefficients of the polynomial   must be zero for it being constantly one,  , contradicting the primality of  .

Thus, let   be maximal containing  . Assume   contains  . Then   and thus  .   contracts to a maximal ideal   of  , which does not contain  , but does contain  . Hence the claim.

The other direction is more tricky, but not as bad as in the previous theorem.

Let thus   be a Jacobson ring. Assume there exists a maximal ideal   such that   is not maximal within  . Define

  and  .   is a prime ideal, since if   such that  ,   or   and hence   or  . Further
 

via the isomorphism

 .

According to lemma 12.5,   is a maximal ideal within  . We set

  and  .

Then   is a Jacobson ring that is not a field,   is a maximal ideal within   (isomorphisms preserve maximal ideals) and  , since if   is any element of   which is not mapped to zero by  , then at least one of   must be nonzero, for, if only  , then  , which is absurd.

Replacing   by   and   by  , we lead the assumption to a contradiction where   is an integral domain but not a field and  .

  is nonzero, because else   would be a field. Let   have minimal degree among the nonzero polynomials of  , and let   be the leading coefficient of  .

Let   be an arbitrary maximal ideal of  .   can not be the zero ideal, for otherwise   would be a field. Hence, let   be nonzero. Since  ,  . Since   is maximal,  . Hence,  , where   and  . Applying the general division algorithm that was described above in order to divide   by   and obtain

 

for suitable   and   such that  . From the equality holding for   we get

 .

Hence,  , and since the degree of   was minimal in  ,  . Since all coefficients of   are contained within   (since they are multiplied by  ),  . Thus   (maximal ideals are prime).

Hence,   is contained in all maximal ideals of  . But since   was assumed to be an integral domain, this is impossible in view of lemma 12.3 applied to the set  , yielding a prime ideal   which is separated from   by a maximal ideal since   is a Jacobson ring. Hence, we have obtained a contradiction.