RICHARD ERWIN HASENAUER

Transcription

1 ALMOST DEDEKIND DOMAINS WITH NONZERO JACOBSON RADICAL AND ATOMICITY RICHARD ERWIN HASENAUER Abstract. We show that if there exists an atomic almost Dedekind domain D with a nonzero Jacobson radical, either D has only finitely many primes or D can be translated into a completely dull domain. That is a domain with only dull primes. Further we show that an atomic dull domain must be wicked, meaning for all nonunits b D we must have N(b) being unbounded. We will then give some properties of wicked domains. A Dedekind domain D with a J(D) 0 can only have finitely many primes. Thus there exists atomic almost Dedekind domains with finitely many maximal ideals that are atomic, they just have to be Dedekind. Throughout the paper we will assume our domain has infinitely many maximal ideals. Taking from [4] we say a maximal ideal is sharp if it is the radical of a finitely generated ideal. Maximal ideals that are not sharp will be called dull. We will call an atom that is contained in only one maximal ideal, M, the sharp atom associated to M. We will make use of previous results. Theorem 0.1. If D is an atomic almost Dedekind domain with J(D) 0, then all sharp primes are principal. Proof. We know from Gilmer in [2], that if M is a sharp prime, there exists m M such that (m) = M. We take m to be the sharp atom associated with M. All we must show is that ν M (m) = 1. Suppose ν M (m) = r > 1. Now we take d 0 J(D), and we find c D with ν M (c) = 1. We construct the element ξ = cd r J(D) and we note that ν M (ξ) = rν M (d) + 1. Now we have m ν M (d) dividing ξ. Furthermore the quoteint ξ t = is in J(D) m ν M (d) and ν M (t) = 1. Thus t + m is only in M and has value 1 on M. Thus the sharp atom must have ν M (m) = 1. Hence for any b M, we have m dividing b and M is principal as claimed. Now we establish a lemma involving the number of sharp primes an atomic almost dedekind domain can have if J(D) 0. Lemma 0.2. If D is an atomic almost Dedekind domain with J(D) 0, then D can only have finitely many sharp primes. Proof. Suppose D has infinitely many sharp primes. Let d 0 J(D). Since D is atomic we factor d into atoms as d = α 1 α 2 α n. We now show that this factorization must be longer. Now α i must be in infinitely many sharp primes for some i. But if α i M i for some sharp prime M i then it must be divisible by m i where m i is the sharp atom associated with M i. Thus this factorization can not possibly be finite. We conclude that D has only finitely many sharp primes. Let {M λ } λ Λ be the set of maximal ideals of D. Recall it was shown in [3] that D is an atomic almost Dedekind domain if and only if Norm(D) = {N(d) = Date: October 10, 2011.

2 (ν Mλ (d)) λ Λ d D} is an atomic moniod. We will refer to the set of maximal ideals that contain an element b as max(b). Lemma 0.3. If D is an atomic almost Dedekind domain with a sharp prime M = (α), then D[α 1 ] is atomic. Proof. We note that α is not in any other maximal ideal, else M would not be maximal. Thus adjoining α 1 only kills of M. Now for b D we can write b = α r β 1 β 2 β n where ν M (b) = r and none of the β i are in M. Thus in D = D[α 1 ] we get transform b into b = β 1 β 2 β n as α becomes a unit. Now we only need to verify that atoms in D remain atoms in D with the only exception being α. If β D is an atom other than α then β / M. Now N(β) is irreducible in the moniod Norm(D). Furthermore N(β) is irreducible in the moniod Norm(D ) for they are the same but with one less entry of zero. Thus if β / M is an atom in D, then it remains an atom in D[α 1 ]. Now we arrive at our conclusion. Theorem 0.4. If D is an atomic almost Dedekind domain with J(D) 0 with infinitely many maximal ideals. Then there exists an atomic dull domain D derived from D. Proof. If D has no sharp primes there is nothing to prove. If D has sharp primes they are all principal and there can only be only finitely many of them. Let (α 1 ), (α 2 ), (α n ) be the list of sharp primes. Now we apply the previous lemma one sharp atom at a time and arrive at D = D[α1 1, α2 1, αn 1 ] is an atomic domain. D has no sharp primes, for we eleminate the ones that existed. If Q is a dull prime of D, it remains dull in D. None of the generators of Q can become units, for α i is not in any dull Q. We now present an example that is quite elegant. An atomic domain is well behaved when it come to factorization. A legitimate dull domain, on the otherhand, is as badly behaved as a domain can be. Recall for [3] that a domain is called legitimate if for all b D we have N(b) being bounded. Glad domains and SP -domains are classes of almost Dedekind domains that satisfy the property of being legitimate. We say a domain D is antimater if D contains no atoms (irreducibles). Another characterization that we will use in the following proof, is that a domian is antimater if for every b D one can find a divisor of b. Theorem 0.5. Let D be a legitimate dull domain with J(D) 0, then D is an antimater domain. Proof. Take b D with {M γ } γ Γ = max(b). We find c J(D) such that b divides c and c is not a multiple of b on Γ. Now as D is legitimate there exists positive integers ρ and π such that for all M we have ρ > ν M (b) and π > ν M (c). Now we consider the set ν M (c) M Max(D) Q. This set is finite, thus it must contain its supremum, say d. We find α Γ such that ν Mα (b) ν Mα (c) = d. That is bα c α = d. Now we consider b cα and c bα. We have for all maximal ideals M, ν M (b cα ) = c α ν M (b) and ν M (c bα ) = b α ν M (c). We claim b cα divides c bα. We verify by

3 observing for all M we have ( ( )) b α ν M (c) c α ν M (b) = ν M (c) b α c α ν M (c) ( ( )) bα ν M (c) b α c α = 0. c α Thus b cα divides c bα and the quotient is in Max(D) \ Λ where Λ Γ. Note the quotient is zero in the α th slot, and is not zero on the entire set of Γ because of our insistance c is not a multiple of b on Γ. Now we have ( ( c b α ) ρ ) { ν M (b) M Λ N b + = b cα 0 M / Λ. And we note ( N b + ( c b α ) ρ ) < N(b). b cα Thus we have found a divisor of b. We conclude that D is an antimater domain. By merely adding the condition of legitimacy, we made a domain with absolutely no atoms. So if D is going to be atomic and dull with J(D) 0, we must have a domain with an element b such that its set of valuations is unbounded, that is the set {ν M (b) M Max(D)} has no upper bound. We will call such elements illegitimate. And we will call a domain with illegitimate elements an illegitimate domain. We note our definition of an illegitimate domain does not prevent the domain from have legitimate elements. Definition 0.6. We call a domain D wicked, if for all b D we have b illegitimate. We will call these wicked domains. We now establish a theorem. Theorem 0.7. If D is an atomic dull domain with J(D) 0, then D is wicked. Proof. Suppose there exists a legitimate element b D. Since we are assuming D is atomic, this implies there must exist a legitimate atom. Thus we will take b to be an atom. Now we find c J(D) such that b c and c is not a multiple of b on Γ = max(b). Now if c were a legitimate element we could find our divisor as we did in the previous theorem. Thus we need to wiggle around the illegitimacy of c, but as we will see this will just involve some observations. Now we are assuming b is legitimate so we find ρ with ρ > ν M (b) for all M Max(D). Our previous proof relied upon finding the supremum of ν M (c) M Max(D) Q. We knew such a supremum existed in the set, becuase the set was finite. But now that we do not have an upper bound on the value of c, the set will be infinite. However, we will see that the set contains its supremum. Let 1 Σ 1 = ν M (c) ν M(b) = 1. Now the set σ 1 = {ν M (c) ν M (b) = 1} is a subset of N 0. We should note that ν M (c) 0 for any M, becuase c J(D). Thus σ 1 contains a least element, say τ 1. Now 1 τ 1 is the supremum of Σ 1. Now for 1 i < ρ we define i Σ i = ν M (c) ν M(b) = i.

4 Again we set σ i = {ν M (c) ν M (b) = i}. We find the least element τ i of σ i and note that τ i i since b c. Now i τ i is the supremum of Σ i. Now since b is bounded in value by ρ we see ( sup 1 ν M (c) M Max(D) = sup(σ 1, Σ 2, Σ ρ 1 ) = sup, 2, ρ 1 ). τ 1 τ 2 τ ρ 1 Thus the supremum exists and is in the set. Now we finish the proof just like in the previous theorem. Thus there must not exist any legitimate elements in D. We conclude that D is a wicked domain. We can make a characterizeation of wicked domains with the following theoreom. Theorem 0.8. For an almost Dedekind domain D the following are equivalent: (i) D is wicked. (ii) For all finite sets {b λ λ Λ} N 0 the ideal is non-principal. λ Λ M b λ Proof. If D is wicked, then D has no legitimate elements. That is if λ Λ M b λ is principal, then {b λ λ Λ} must be unbounded. Conversely if or all finite sets {b λ λ Λ} the ideal λ Λ M b λ is not principal, then D has no legitimate elements. Thus D is wicked. Now wicked domains have some properties that we will refer to as the LEG lemmas. First we define what is meant by LEG. Let a, b be elements of a wicked domain D We define the symmetrically related sets: L = {M Max(D) ν M (a) < ν M (b)}, E = {M Max(D) ν M (a) = ν M (b)}, G = {M Max(D) ν M (a) > ν M (b)}. Lemma 0.9. If D is a wicked domain with J(D) 0. Then for a, b D with ab J(D) we must have either L = {ν M (a) M L} or G = {ν M (b) M G} being unbounded or E being infinite. Proof. Suppose L and G are both bounded and E is finite. Then ν M (a) M L N(a + b) = ν M (b) M G M E a i Now we must have a + b being a legitimate element which is inpossible. Thus the result must be true. Lemma Suppose α, β are atoms in a wicked domain D such that αβ J(D), then E. Proof. Suppose E = Then { N(α + β) = ν M (α) ν M (β) M L M G But now N(α + β) < N(α) and (α + β) α. But α was an atom, thus we must have E.

5 Note if we are careful we can remove the restriction of β being an atom. We would just need to make sure that α does not divide β, which of course would mean that both would be in J(D). We now present an approximation theorem. It should be noted that the approximation theorem is true for any almost Dedekind domain. Theorem Let D be an almost Dedekind domain. If S Max(D) such that M S M J(D) then there exists a nonunit b D such that ν M (b) = 0 for all M S. Proof. We find b M S M \ J(D). Now since b / J(D), there exist r D such that 1 rb is not a unit. Now rb M for all M S. Hence 1 rb / M for all M S, hence ν M (1 rb) = 0 for all M S. Corollary Let D be an almost dedekind domain and let a J(D). For n N set S n = {M ν M (a) n}. If D is wicked, then M Sn M = J(D). Proof. Suppose for a given n we have M Sn M J(D). Then there exists b D such that ν M (b) = 0 for all M S n. Now we have { N(b n 0 M S n + a) = < n M / S n. But then b n + a is a legitimate element, hence D is not wicked. Finally we end with one last characterization of wicked. Lemma If D is wicked then there do not exist nonunital elements a, b in D such that ab J(D) and max(a) max(b)=. Proof. Suppose to the contrary there such and a, b D. Now set S n = {M ν M (a) n}. We know that M Sn M J(D) since it contains a and a / J(D). Thus we can find nonunit c with ν M (c) = 0 for M S n. But now consider ξ = (bc) n+1 + a. We see that ξ has a norm bounded by n and is legitimate. Thus we have a contradiction and the lemma must be true. We make a note as a corollary. Corollary A wicked domain D can not be contructed via integral extension from a domian with one prime. The existence of wicked domains has perplexed the author. It is conjectured that no such domains can exist and be almost Dedekind. Regardless of whether the conjecture is true, it would be nice to see the question resolved. If there does exist an almost Dedekind wicked domain can it be atomic? Perhaps this question might be easier to answer. References [1] R. Gilmer, Multiplicative Ideal Theory Queen s papers Pure Appl. Math. 90, Queen s University Press, Kingston 1992 [2] R. Gilmer, Overrings of Prüfer domains, J. Algebra 4 (1966), [3] R. E. Hasenauer Normsets of almost Dedekind domains and atomicity, preprint [4] K.A. Loper and T.G. Lucas, Factoring ideals in almost Dedekind domains, J. reine angew. Math., 565, (2003), [5] K.A. Loper, More almost Dedekind domains and Prúfer domains of polynomials, in Zerodimensional commutative rings (Knoxville, TN, 1994), , Lecture Notes in Pure and Appl. Math., 171 (1995), Dekker, New York. [6] B. Olberding, Factorization into radical ideals, Lec. Notes Pure Appl. 241 (2005),