aa r X i v : . [ m a t h . L O ] J un DP-MINIMAL INTEGRAL DOMAINS
CHRISTIAN D’ELB´EE † AND YATIR HALEVI ∗ Abstract.
It is shown that every dp-minimal integral domain R is a localring and for every non-maximal prime ideal p of R , the localization R p is avaluation ring and p R p = p . Furthermore, a dp-minimal integral domain isa valuation ring if and only if its residue field is infinite or its residue field isfinite and its maximal ideal is principal. Introduction
Model theory has a long and fruitful history of classifying algebraic structuresunder some model theoretic constraints. For example, every ω -stable infinite field isalgebraically closed, see [Mac71]. This result was generalized by Cherlin-Shelah tosuperstable division rings in [CS80]. Recently, stable division rings of finite dp-rankwere also shown to be algebraically closed by Palac´ın and the second author [HP19].The end goal of this line of results is the long standing conjecture that every stablefield is separably closed. Although there are no example of stable infinite divisionrings which are not fields, a positive answer to this conjecture would imply thatstable division rings are fields [Mil19, Remark 5.4].Classifying fields failing to have the independence property (i.e., NIP fields) hashad some very encouraging recent results. Starting with the result by Johnson thatany dp-minimal infinite field is either algebraically closed, real closed or admits anon-trivial definable henselian valuation [Joh18], and very recently a generalizationof this result by Johnson to fields of finite dp-rank [Joh20], which encompasses theresult in [HP19]. As a generalization of the stable fields conjecture, it is conjecturedthat every infinite NIP field is either separably closed, real closed or admits adefinable henselian valuation.As for rings, every ( λ -)stable commutative ring with identity can be decomposedin a unique way as a product of R × · · · × R k of local rings ( R i , M i ) such that M i is nilpotent and R i / M i is a ( λ -)stable field [CR76]. There are many other resultsalong this alley, but the moment one assumes that the ring is a stable integraldomain we can easily conclude, by DCC on principal ideals, that the ring is adivision ring which brings us back to the realm of the first paragraph.Remaining on the subject of rings but relaxing the model theoretic assumptionsfrom stable to NIP , Milliet has shown recently that every NIP division ring ofpositive characteristic has finite dimension over its center and that there are non-commutative NIP division rings of every characteristic [Mil19]. Hempel-Palac´ınhave shown that any division ring of burden n has dimension at most n over itscenter, hence dp-minimal division rings are fields [HP18]. Dobrowolski-Wagner haveshown that every ω -categorical ring of finite dp-rank is virtually null [DW20]. Onthe other hand, NIP integral domains that are not valuation rings have not received † Supported by ISF grant No. 1254/18. ∗ Partially supported by ISF grant No. 181/16 and the Kreitman foundation fellowship. There are other results with different model theoretic assumptions, e.g. every O-minimaldomain is a division ring and in fact it is RCF, ACF or the quaternions [OPP96, PS99]. much attention. The main focus of this paper is dp-minimal integral domains andspecifically their connection to valuation rings.We show that the prime spectrum of an inp-minimal integral domain is linearlyordered by inclusion, and hence a local ring (Corollary 2.4). A dp-minimal domainis much closer to a valuation ring: every non-maximal prime ideal is the maximalideal of a valuation overring. The main results are summarized in the following.
Theorem.
Let R be a dp-minimal integral domain with maximal ideal M .(1) R p is a valuation ring for every non-maximal prime ideal p ;(2) R is a divided ring, in the sense that p R p = p for every prime ideal p ;(3) R is a valuation ring if and only if one of the following holds(a) R/ M is infinite;(b) R/ M is finite and M is a principal ideal.(4) if K = Frac( R ) then for every externally definable valuation subring O of K , either O ⊆ R or R ⊆ O .(1) is proved in Proposition 3.8 after we show that R p with a predicate for R is still dp-minimal for a large enough family of prime ideals p (Proposition 2.8). (2) is proved in Theorem 3.9. (3) and some other connected results are shown inTheorem 4.1 and (4) is Corollary 5.6.Regarding divided rings, they were first studied by Akiba, who used the term“AV-domains” [Aki67]. Later, Dobbs studied them when searching for an internalcharacterization for going-down domains (domains R for which R ⊆ S is goingdown for every overring S ). Dobbs proved that if R a root-closed domain then R is going-down if and only if R is divided, see [Dob76] for more information. Acknowledgments.
We would like to thank Will Johnson for the idea behind Propo-sition 2.8, Eran Alouf for Lemma 3.6(1), Philip Dittman and Itay Kaplan for usefuldiscussions. Finally, we would like to thank Pierre Simon for suggesting this project.1.1.
Notation and Preliminaries.
A ring will always mean a commutative ringwith identity and a domain is an integral domain. For any a ∈ R , the ideal generatedby a will be denoted by aR , or h a i if it is unambiguous. Let char( R ) denote thecharacteristic of the ring R . We denote by Frac( R ) the fraction field of a domain R and an overring of R is a ring R ⊆ S ⊆ Frac( R ). Model theoretic notation andnotions are standard, see for example [TZ12] and [Sim15].2. General Results
We start with some general results. The following was already observed by PierreSimon.
Proposition 2.1.
Let R be an NIP ring and ( p i ) i<ω an infinite family of primeideals. Then there is i < ω such that p i ⊆ S j = i p j . Equivalently, there is noinfinite antichain of prime ideals and in particular, there is only a finite number ofmaximal ideals.Proof. Assume otherwise and let a i ∈ p i \ S j = i p j , and for each finite I ⊆ ω , set b I = Q i ∈ I a i . Let φ ( x, y ) be the formula y ∈ h x i , i.e. ∃ z ( y = zx ). Then for every i < ω and finite set I ⊆ ω , φ ( a i , b I ) holds if and only if i ∈ I . Indeed, if i ∈ I thenit is clear that b I ∈ h a i i . Since the ideals p i are prime, b I ∈ T i ∈ I p i \ ( S j / ∈ I p j ) soif i / ∈ I then b I / ∈ h a i i . (cid:3) Remark . The proof of the previous proposition shows that in any commutativering, the maximal length of an antichain of prime ideals is bounded by the VC-dimension of the formula y ∈ h x i . P-MINIMAL INTEGRAL DOMAINS 3
For domains of finite burden, the burden also bounds the maximal length of anantichain of prime ideals:
Proposition 2.3.
Let R be an integral domain of burden n ∈ N and let p , . . . , p n +1 be prime ideals of R . Then there exists ≤ i ≤ n + 1 such that p i ⊆ S j = i p j .Proof. Assume not, then for each 1 ≤ i ≤ n + 1 there exists a i ∈ p i \ S j = i p j . Also,since the ideals are prime, a ki ∈ p i \ S j = i p j for all k ≥
1. For each 1 ≤ i ≤ n + 1,and k ≥
1, let X ki be the set h a ki i \ h a k +1 i i . The latter is nonempty: assume that a ki ∈ h a k +1 i i , then for some b ∈ R , a ki = a k +1 i b . Since R is an integral domain, itfollows that a i is a unit in R , which contradicts that p i is an ideal.We now conclude that { x ∈ X ki } ≤ i ≤ n +1 ,k ≥ is an inp-pattern of length n + 1.Let k , . . . , k n +1 ≥
1. We claim that a k · . . . · a k n +1 n +1 ∈ X k ∩ · · · ∩ X k n n . Indeed,if, without loss of generality, a k · . . . · a k n +1 n +1 = a k +11 b , for some b ∈ R , then a k · . . . · a k n +1 n +1 ∈ h a i ⊆ p . Consequently, a j ∈ p for some j = 1, contradicting thechoice of the a i . To complete the argument, note that the rows are 2-inconsistent:as before, since R is an integral domain if 1 ≤ i ≤ n + 1 and s = t ≥ X si ∩ X ti = ∅ . (cid:3) Corollary 2.4.
In an inp-minimal domain the prime spectrum is linearly orderedby inclusion. In particular, all the proper radical ideals are prime and there exists N ∈ N such that for all a, b ∈ R either b N ∈ h a i or a N ∈ h b i .Proof. The fact that the prime ideals are linearly ordered follows directly from theproposition. The fact that all radical ideals are prime follows from the fact that theradical of an ideal is the intersection of all prime ideals containing it. Let a, b ∈ R then either p h a i ⊆ p h b i or p h b i ⊆ p h a i , so either a n ∈ h b i or b n ∈ h a i for some n ∈ N . To find a uniform N we use compactness. (cid:3) Remark . A domain with a linearly ordered prime spectrum is also called in theliterature a local treed domain [DDAa00, Chapter 7] [Bad95].
Corollary 2.6.
Let R be an integral domain of burden n , then R has at most n maximal ideals. If R is an inp-minimal integral domain which is not a field, then R has exactly one maximal ideal, i.e. R is a local ring. Fields have received more attention in model theory than domains. It would bebeneficial to be able to localize without compromising some of the properties of thetheory. For that we will require the following lemma.
Lemma 2.7.
Let R be an integral domain and S a multiplicatively closed subset of R . Then for every formula ϕ (¯ x, c ) in ( S − R ; R, S, + , · , , , with c ∈ S − R thereexists a formula ϕ ∗ (¯ x, c ∗ ) in ( R ; S, + , · , , , with c ∗ ∈ R such that { ¯ a ∈ R : ( R ; S ) | = ϕ ∗ (¯ a, c ∗ ) } = { ¯ a ∈ S − R : ( S − R ; R, S ) | = ϕ (¯ a, c ) ∧ R (¯ a ) } . Proof.
We first prove the following: for every such formula ϕ ( x , . . . , x n , c ) we canfind a formula ϕ † ( x , y , . . . , x n , y n , c ∗ ), with bounded quantifiers (i.e. quantifiersof the form ∃ u ∈ R or ∃ u ∈ S ) and c ∗ ∈ R such that { ( a /b , . . . , a n /b n ) : ( S − R ; R, S ) | = ϕ † ( a , b , . . . , a n , b n , c ∗ ) , a i ∈ R, b i ∈ S } = { ¯ a ∈ S − R : ( S − R ; R, S ) | = ϕ (¯ a, c ) } . We prove it by induction on the complexity of ϕ . Assume ϕ ( x, c ) is atomic. If itis of the form S ( P (¯ x )) (or R ( P (¯ x ))) for some polynomial P (¯ x ) then we consider theformula ( ∃ x n +1 )( x n +1 = P (¯ x ) ∧ S ( x n +1 )), (or ( ∃ x n +1 )( x n +1 = P (¯ x ) ∧ R ( x n +1 ))and use the induction hypothesis. As a result, we may assume it is of the form R ( x ) or S ( x ) or P (¯ x ) = 0 for some polynomial P over Z [ c ]. If it is of the form C. D’ELB´EE AND Y. HALEVI R ( x ), the corresponding formula is ( ∃ z ∈ R )( zy = x ) and likewise for S ( x ).Assume ϕ ( x, c ) is of the form P (¯ x ) = X i ,...,i n c i ,...,i n x i . . . x i n n = 0 . Then, consider the formula obtained by replacing each x i by x i /y i and clear de-nominators (from the coefficients as well), i.e. multiplying the whole equation bythe product of all the y i ’s and the denominators of the c i ,...,i n .Conjunctions are straightforward and so are negations. If ϕ (¯ x, c ) is of the form( ∃ x n +1 ) ψ ( x , . . . , x n , x n +1 , c ) then the corresponding formula is( ∃ x n +1 ∈ R, y n +1 ∈ S ) ψ † ( x , y , . . . , x n +1 , y n +1 , c ∗ ) . Finally we define ϕ ∗ (¯ x, c ∗ ) to be ϕ † ( x , , . . . , x n , , c ∗ ) (after replacing all in-stances of R ( u ) by u , for any variable u ). (cid:3) Proposition 2.8.
Let R be an integral domain and S a multiplicatively closedsubset of R .(1) If S is definable then the burden of R is equal to then burden of ( S − R, R ) .In particular if R is NTP then so is ( S − R, R ) .(2) If S is externally definable in R and R is NIP then ( S − R, R ) is NIP and asa result, by (1) , dp-rk ( R ) < κ ⇐⇒ dp-rk ( S − R, R ) < κ , for any cardinal κ .Proof. (1) Since S is definable there is no harm in adding it as a predicate. Since R is definable in ( S − R, R ) one direction is obvious. For the other direction, let N be a monster model and assume that { ϕ i ( x, y i ) , ( a i,j ) j<ω , k i } i<κ , where a i,j ∈ N , is an inp-pattern of depth κ in ( S − R ; R, S ), i.e. • for all i < κ , the i-th row is k i inconsistent • for every f : κ → ω , { ϕ i ( x, a i,f ( i ) } i<κ is consistent, witnessed by b f .Since, in ( S − R ; R, S ), for any finite set A ⊆ S − R there exists 0 = c ∈ R with c · A ⊆ R , we may find a nonzero element c ∈ R ( N ) such that c · a i,j ∈ R ( N ) forall i < κ and j < ω and c · b f ∈ R ( N ) for all f : κ → ω .Now, for every i < κ let ϕ ∗ i ( x, y, c ∗ i ) be the formula corresponding to ϕ i ( x · c − , y · c − ) as supplied by Lemma 2.7. It is now routine to check that by Lemma2.7 { ϕ ∗ i ( x, y, c ∗ i ) , ( ca i,j ) j<ω , k i ) } i<κ is an inp-pattern of depth κ as witnessed by ( cb f ) f : κ → ω . (2) Since S is externally definable, by [Sim15, Corollary 3.24] there is no harm inadding S to the language. The result follows since ( S − R, S ) is interpretable in(
R, S ). (cid:3) Corollary 2.9.
Let R be valuation ring. If either R is dp-minimal or char ( R ) = p and R is NIP then R is henselian.Proof. By [JSW17, Proposition 4.5] (or [Joh16, Corollary 9.4.16]) every dp-minimalvalued field is henselian and by [Joh19, Theorem 2.8] every NIP valued field ofpositive characteristic is henselian. As a result the statement will follow by applyingthe Proposition with S = R \ { } . (cid:3) To end the section, we note the following, which is just a rephrasing of thecorresponding statement for dp-minimal fields.
Lemma 2.10.
Let R be an inp-minimal ring. P-MINIMAL INTEGRAL DOMAINS 5 (1) For any two definable sets X and Y , if X and Y are infinite then for all a ∈ R \ { } , a ( Y − Y ) ∩ ( X − X ) = { } . In particular, if R is an integraldomain then R eliminates ∃ ∞ .(2) R is a field if and only if there exists an infinite definable set X ⊆ R suchthat X − X ⊆ R × ∪ { } . In particular R admits a definable infinite subringwhich is a field if and only if R is itself a field.Proof. (1) Otherwise, the function ( x, y ) x + ay would be a definable injectionfrom X × Y into R , contradicting that R is inp-minimal.We show that for integral domains, a definable set X is infinite if and only if forall a ∈ R , a ( X − X ) ∩ ( X − X ) = { } . The “only if” direction is the previous proof.For the “if”, let K = Frac( R ). If R is finite, then the result holds trivially. Assumethat R is infinite, X is finite, but that for all a ∈ R \{ } , a ( X − X ) ∩ ( X − X ) = { } ,then R is included in { a/b | a ∈ X − X, b ∈ ( X − X ) \ { }} ⊆ K which is a finiteset. (2) If R is a field we may take X = R \ { } . For the other direction, assume thatsuch an infinite definable set X exists. Let a ∈ R \ { } and consider the function X × X → R mapping ( x, y ) ax + y . Since R is inp-minimal, the map cannot beinjective. Hence there exists u ∈ X − X ⊆ R × with a = u so a is invertible. (cid:3) Algebraic Properties of Dp-Minimal Domains
Let R be a dp-minimal domain. In Corollary 2.4 we have seen that there exists N ∈ N such that for all a, b ∈ R either a N ∈ h b i or b N ∈ h a i . A connected notionis that of a divided domain. Definition 3.1. ([Aki67],[Dob76]) A prime ideal p of a ring R is called divided iffor all a ∈ R we have p ⊆ h a i or h a i ⊆ p . Equivalently, a prime ideal p is divided ifand only if p = p R p . A ring R is divided if every prime ideal of R is divided. Remark . It is not hard to see that a ring R is divided if and only if for any a, b ∈ R either a ∈ h b i or b n ∈ h a i for some n ∈ N . If p is a divided prime idealthen it is also comparable to every ideal.In this section we prove that every dp-minimal ring is divided. There is no hopeto assume in general that all prime ideals are (externally) definable. Nevertheless,as the following will show, all prime ideal are an intersection of externally definableprime ideals. Lemma 3.3.
Let R be an inp-minimal domain with maximal ideal M .(1) For each a ∈ R there exists a unique ideal P a such that P a is maximal withthe property P a ∩ { a n | n ∈ N } = ∅ . P a is prime and externally definable.(2) If R is dp-minimal then ( R P a , R ) is dp-minimal in the language of ringswith a predicate for R .(3) For any prime ideal p , p = T a ∈ R \ p P a .Proof. (1) Let a ∈ R . By Zorn’s lemma we may find a maximal ideal P satisfyingthat P ∩ { a n | n ∈ N } = ∅ . Assume that P is not a prime ideal, so there exists x, y ∈ R such that xy ∈ P and x / ∈ P , y / ∈ P . By maximality, P + h x i and P + h y i intersect { a n | n ∈ N } , i.e. there exists c , c ∈ P and r , r ∈ R such that c + r x and c + r y are in { a n | n ∈ N } . It follows that the product ( c + r x )( c + r y ) isin P ∩ { a n | n ∈ N } a contradiction. Uniqueness follows from Corollary 2.4.We show that R \ P is externally definable. Consider the uniformly definablefamily of sets ϕ ( x, a n ) := a n ∈ h x i . The result will follow from [HHJ19, Lemma3.4] once we establish that [ n ∈ N ϕ ( R, a n ) = R \ P. C. D’ELB´EE AND Y. HALEVI
Let b ∈ R \ P , if we would have h b i∩{ a n : n ∈ N } = ∅ , then h b i would be contained insome maximal ideal satisfying this property, by uniqueness h b i ⊆ P , contradiction.For the other direction, if a n ∈ h b i , then we cannot have that b ∈ P . (2) By Proposition 2.8(2) and (1) the structure ( R P a , R ) is dp-minimal (in thering language with a predicate for R ). (3) Let a / ∈ p . By the definition of P a , p ⊆ P a . For the other direction, let b ∈ T a/ ∈ p P a . If b / ∈ p then b ∈ P b , contradicting the definition of P b . (cid:3) Remark . Note that for any a ∈ R , R P a = S − R , where S = { a n | n ∈ N } . Theideals of the form P a are the so-called Goldman ideals in the literature (see forinstance [Pic76]). Fact 3.5. [Sim15, Proposition 4.31]
Let G be a inp-minimal group and H, N de-finable subgroups. Then either | H/H ∩ N | < ∞ or | N/H ∩ N | < ∞ . Lemma 3.6.
Let R be an inp-minimal integral domain with maximal ideal M . If R contains an infinite set F such that F − F ⊆ R × ∪ { } then R is a valuationring. In particular,(1) if R/ M is infinite then R is a valuation ring;(2) if R has an infinite subring which is a field then R is a valuation ring.Proof. Assume that such a set F exists, and let ( f i ) i<ω ⊆ F be such that f i = f j forall i = j . Let a, b ∈ R be nonzero elements. Then by Fact 3.5, either h a i / ( h a i∩h b i ) isfinite or h b i / ( h a i∩h b i ) is finite. Without loss of generality, assume that h a i / ( h a i∩h b i )is finite. As ( f i a ) i ⊆ h a i , it follows that there exists i = j such that ( f i a − f j a ) ∈ h b i .As F − F ⊆ R × ∪ { } , we conclude that a ∈ h b i . (cid:3) Remark . In a local ring, every non-maximal radical ideal has infinite index inthe maximal ideal M (as additive groups). Indeed, if r ( M is a radical ideal thenfor any b ∈ M \ r and n = m ∈ N , b n and b m are in different classes modulo r .As a result we get the following, which is almost the definition of a Pr¨ufer domain. Proposition 3.8.
Let R be a dp-minimal domain with maximal ideal M and p anon-maximal prime ideal. Then R p is a (possibly trivial) henselian valuation ring.Proof. Since p is non-maximal, by Lemma 3.3(2) and since the prime ideal arelinearly ordered by Corollary 2.4 there exists some a ∈ M \ p with p ⊆ P a ( M .This implies that R P a ⊆ R p and as a result it is enough to show that R P a is avaluation ring.We note that since | R/ p | ≤ | R P a /P a R P a | , the maximal ideal P a R P a of R P a hasinfinite index. Note that ( R P a , R ) is dp-minimal by Lemma 3.3(2). By Lemma 3.6, R P a (and thus also R p ) is valuation ring. It is henselian by Corollary 2.9. (cid:3) Theorem 3.9.
Every dp-minimal domain is divided. In particular, there exists N ∈ N such that for all a, b in the maximal ideal, either a ∈ h b i or b N ∈ h a i .Proof. Let R be a dp-minimal ring with maximal ideal M and let p be a primeideal. Since R is a local ring, the maximal ideal is divided and so we may assumethat p ( M . Since R ∩ p R p = p , it will be enough to show that p R p ⊆ R . ByLemma 3.3(3), there exists some a ∈ M \ p such that p R p ⊆ P a R P a and hence itis enough to show that P a is divided. Set P := P a . By Lemma 3.3(2), ( R P , R ) isdp-minimal.As P R P and R are two definable (additive) subgroups in this structure, by Fact3.5, either | R/P | < ∞ or | P R P /P | < ∞ . Since P necessarily has infinite index in R it must be the latter. Let y , . . . , y n be representatives for the different cosets of P in P R P . In particular { , y , . . . , y n } generate R + P R P as an R -module, i.e. itis a finitely generated R -module. P-MINIMAL INTEGRAL DOMAINS 7
Either by direct computation or by dp-minimality, it is not hard to see that R + P R P is a local ring and its maximal ideal is M + P R P . Consider the dp-minimal ring R + P R P as an R -module. Since M ( R + P R P ) = M + P R P and( R + P R P ) / ( M + P R P ) ∼ = R/ M , by Nakayama’s lemma the generator 1 + ( M + P R P ) of ( R + P R P ) / ( M + P R P ) lifts to a generator of R + P R P as an R -module,i.e. R = R + P R P , as needed.The “in particular” follows by compactness and Remark 3.2. (cid:3) Remark . Overrings of the form R + P R P are called CPI-extensions of R [BS77].We learned the Nakayama trick for CPI-extensions from [Dob78, Lemma 2.3]. Corollary 3.11.
Let R be a dp-minimal domain. Then R is a (henselian) valuationring if and only if there exists a non-maximal prime ideal p such that R/ p is avaluation ring.In particular, if the maximal ideal of R is principal and R has a finite residuefield then R is a valuation ring.Proof. If R is a valuation ring and p is any non-maximal ideal then R/ p is also avaluation ring.For the other direction, assume that p is a non-maximal prime ideal and that R/ p is a valuation ring. By Lemma 3.8, R p is a valuation ring. By definition, R + p R p is the composition of the valuation ring R p and the valuation ring R/ p ∼ =( R + p R p ) / p R p of the field R p / p R p (see also [BS77, page 724]) and hence a valuationring as well. Since R is divided, R + p R p = R and we conclude that R is a valuationring.Let M be the maximal ideal of R , assume that M = h π i is principal and that R/ M is finite. Let P π be the maximal ideal that does not intersect { π, π , . . . } .By definition, for all 0 = a ∈ R/P π , π n + P π ∈ aR/P π for some n . Since R/ M isfinite, it follows that aR/P π is of finite index in R/P π . Consequently, R/P π is aNoetherian local domain with principal maximal ideal, hence a valuation ring. (cid:3) Let R be any integral domain, K its fraction field and let I be an ideal. Wedenote I : I = { x ∈ K | xI ⊆ I } and I − = { x ∈ K | xI ⊆ R } . Corollary 3.12.
Let R be a dp-minimal domain and p a non-maximal prime idealthen p : p = R p . If M is the maximal ideal and R is not a valuation ring then M − = M : M . Proof.
The first assertion is a consequence of Theorem 3.9, Proposition 3.8 and[Oka84, Lemma 6].Let M be the maximal ideal of R and assume that R is not a valuation ring. ByCorollary 3.11, M is not principal. Assume there exists x ∈ K with x M ∩ ( R \ M ) = ∅ so x − R = M . This implies that M is a principal ideal of R , contradicting ourassumption. (cid:3) For a (type-)definable group G in an NIP structure we denote by G the smallesttype-definable subgroup of G of bounded index, for more information see [Sim15,Section 8.1.3]. Fact 3.13. [Joh19, The proof of Lemma 2.6]
Let R be an NIP integral domain and I a definable ideal. Then I is also an ideal. C. D’ELB´EE AND Y. HALEVI
Proposition 3.14.
Let R be a dp-minimal domain with maximal ideal M andfraction field K . Then(1) M : M is a valuation overring of R ;(2) √ M = { a ∈ M | M / h a i is infinite } ;(3) For every a ∈ R with M / h a i is finite, √ M = P a .In particular, exactly one of the following holds: • M / h a i is infinite for all a ∈ M and √ M = M ; • √ M = P a for any a ∈ M with M / h a i finite.Remark . It follows that M = M if and only if M is prime and M / h a i isinfinite for all a ∈ M . Proof. (1)
It is routine to show that M : M is a ring, so it is sufficient to provethat it is a valuation ring, i.e. for all a ∈ K , a M ⊆ M or a − M ⊆ M . Let a ∈ K . By [KS13, Proposition 3.12], a M / ( M ∩ a M ) is finite or M / ( M ∩ a M ). By multiplying by a − we may assume without loss of generality that itis the latter. Since M has no finite index type-definable subgroups, M = M ∩ a M , i.e. a − M ⊆ M . (2) We show that a ∈ M \ √ M if and only if R/ h a i is finite. Let a / ∈ √ M .By [KS13, Proposition 3.12], h a i / ( M ∩ h a i ) is finite or M / ( M ∩ h a i ) is finite.If it was the former, we would have that a n ∈ M for some n , i.e. a ∈ √ M ,a contradiction. Consequently, M / ( M ∩ h a i ) is finite and by the definition of M , M ⊆ h a i hence R/ h a i is finite.Conversely, if R/ h a i is finite, then as | R/a M | = | R/ h a i||h a i /a M | = | R/ h a i|| R/ M | , it follows that M ⊆ a M . As M is an ideal we have M = a M . Assumethat a n ∈ M with n is minimal. Thus a n = am for some m ∈ M , but R is adomain so a n − ∈ M , contradiction. (3) Assume that M / h a i is finite. As R is divided, P a ⊆ h a i . In fact, it followsfrom the definition that P a = T n h a n i , hence P a is of bounded index in M (sinceeach h a n i is of finite index in M ). As a result, M ⊆ P a and hence √ M ⊆ P a .For the other inclusion, if b ∈ P a then M / h b i is infinite by Remark 3.7 and so, by (2) , b ∈ √ M . (cid:3) We end with the following.
Definition 3.16 ([HH78]) . An ideal p of a domain R is strongly prime if for all x, y ∈ Frac( R ), xy ∈ p implies x ∈ p or y ∈ p . A domain is a pseudo-valuationdomain if every prime ideal is strongly prime.Every valuation ring is a pseudo-valuation domain, every pseudo-valuation do-main is divided, and a local ring ( R, M ) is a pseudo-valuation domain if and onlyif M is strongly prime [HH78, Theorem 1.4]. Proposition 3.17.
Let R be a dp-minimal domain, then every non-maximal primeideal is strongly prime.Proof. By Proposition 3.8, if p is a non-maximal prime ideal of R , R p is a valuationring, so in particular it is a pseudo-valuation domain and so p R p is strongly prime.As R is divided, by Theorem 3.9, p = p R p , so p is strongly prime. (cid:3) Remark . Dp-minimal domains are not far from being pseudo-valuation do-mains. However, it is not hard to check that F p + F p t + { x | v ( x ) ≥ } is a dp-minimal subring of the valued field ( F algp (( t Q )) , v ), where v is the natural valuation,which is not a pseudo-valuation domain. See Example 4.2, for an explanation why P-MINIMAL INTEGRAL DOMAINS 9 the valued field is dp-minimal. Proposition 6.10 gives a sufficient condition for adp-minimal domain to be a pseudo-valuation domain.4.
Valuation Rings
It is well known (and details will be given below) that every dp-minimal valuedfield of equicharacteristic has an infinite residue field and that every dp-minimalvalued field of mixed characteristic with finite residue field is of finite ramification.We will show below that these are the only obstructions for a dp-minimal ring tobe a valuation ring.Let R be an integral domain with fraction field K . R is root-closed if for all q ∈ K such that q n ∈ R for some n ≥ q ∈ R . It is standard that valuationrings are integrally closed and hence root-closed. Theorem 4.1.
Let R be an infinite dp-minimal integral domain.(1) Assume that R is of equicharacteristic p ≥ . The following are equivalent.(a) R is a henselian valuation ring;(b) R is integrally closed;(c) R is root-closed;(d) R has an infinite residue field;(e) R has an infinite subring which is a field (necessarily Q or F algp );(2) Assume that R is of mixed characteristic (0 , p ) with maximal ideal M . Thefollowing are equivalent.(i) R is a henselian valuation ring;(ii) Either R has an infinite residue field or M is principal (and the residuefield is finite).Proof. (a) implies (b) is standard and (b) implies (c) is clear. (c) implies (d) is clear in equicharacteristic 0. Assume that R is of equicharac-teristic p >
0. By Proposition 2.8, the fraction field K = Frac( R ) is dp-minimaland since it is infinite (as R is infinite), by [KSW11, Corollary 4.5], F algp ⊆ K . Asevery element in F algp is an n -th root of an element of F p ⊆ R for some n , it followsthat F algp ⊆ R . By Lemma 3.6, R is a valuation ring. It follows from [KSW11,Proposition 5.3] that the residue field is infinite. (d) implies (a) is Lemma 3.6. (a) implies (e) . Every local ring of equicharacteristic 0 contains Q . In equi-characteristic p >
0, this is contained in the proof of (c) implies (d) . (e) implies (a) is Lemma 3.6. (i) implies (ii) . By Proposition 2.8, ( K, R ) is dp-minimal, where K = Frac( R ).Let v be the valuation R induces on K . If the residue field is finite, by [Joh16,Theorem 4.3.1], [0 , v ( p )] is finite. In particular, M is principal. (ii) implies (i) . If R has an infinite residue field then R is a valuation ring byLemma 3.6. If R has a finite residue field and M is principal then R is a valuationring by Corollary 3.11. (cid:3) In equicharacteristic 0, the condition ( d ) of Theorem 4.1 is vacuously true, henceevery dp-minimal integral domain of equicharacteristic 0 is a valuation ring. How-ever there are dp-minimal domains of equicharacteristic p > Example 4.2.
Consider K = F algp (( t Q )), the Hahn series over F algp with valuegroup Q , together with the natural valuation v . It is a dp-minimal valued field bye.g. [Joh16, Theorem 9.8.1]. Consider F p + { x | v ( x ) ≥ } , it is a definable subringof K and hence dp-minimal. It is of equicharacteristic p > F p t + { x ∈ K | v ( x ) ≥ } and { x ∈ K | v ( x ) > } are incomparable. Note that thesame argument gives that any ring of the form F p + I is not a valuation ring, forany ideal I of the valuation ring of F algp (( t Q )). Example 4.3.
Let ( Q p , v ) be the p-adics numbers for p = 2, it is dp-minimalby [DGL11, Theorem 6.13], and let K := Q p ( √ p ) be the totally ramified finiteextension given by adjoining the square root of p , it is also dp-minimal (togetherwith the valuation v ). The ring R := { , . . . , p − } + { x ∈ K : v ( x ) ≥ } isdefinable and hence dp-minimal. Note that the maximal ideal is not principal, h p i is not the maximal ideal since it does not contain p √ p . In fact, the maximal idealis generated by p and p √ p . This shows that it is not a valuation ring and that fordp-minimal rings, R/pR finite does not imply a principal maximal ideal.
Remark . Let O p be the valuation ring of the valued field F algp (( t Γ )), for somedivisible ordered abelian group Γ. Let I p any ideal of O p , then R p := F p + I p is nota valuation ring (see Example 4.2). Let U be a non-principal ultrafilter on the setof prime numbers. The ultraproduct Q U O p is dp-minimal since it is the valuationring of an algebraically closed valued field, however Q U R p is not even inp-minimal.Indeed, it is not a valuation ring (as none of the R p are), but it has a pseudo-finite–hence infinite– residue field. If Q U R p were inp-minimal, this would contradictLemma 3.6. 5. Externally Definable Valuation Rings
It is known that every sufficiently saturated non-algebraically closed dp-minimalfield admits an externally definable valuation, [Joh18, Theorem 1.5]. In this section,given a dp-minimal domain, we describe the interactions between R and externallydefinable valuation rings of Frac( R ). Lemma 5.1.
Let R be a local domain and O a valuation ring of Frac( R ) . Then R/ ( O ∩ R ) is finite if and only if R ⊆ O .Proof. It is clear that R/ ( O ∩ R ) is finite if R ⊆ O . Let M be the maximalideal of R and m the maximal ideal of O . Assume, towards a contradiction, that | R/ ( O ∩ R ) | = n ≥
2. There exists some a , . . . , a n − ∈ R \ O (so a − i ∈ m ) suchthat R = ( O ∩ R ) ∪ n − [ i =1 (( O ∩ R ) + a i ) ( ⋆ ) . If a ∈ R × , then a − ∈ R ∩ O and by multiplying ( ⋆ ) by a − we get: a − R = R = a − ( O ∩ R ) ∪ ( a − ( O ∩ R ) + 1) ∪ n − [ i =2 ( a − ( O ∩ R ) + a − a i ) . Since a − O ⊆ O and ( a − O + 1) ⊆ O , this implies that R = ( O ∩ R ) ∪ n − [ i =2 (( O ∩ R ) + a − a i ) . contradicting that the index is n ,If a / ∈ R × then a ∈ M hence 1 + a , (1 + a ) − ∈ R × . Now translating ( ⋆ ) by1, we get R + 1 = R = (( O ∩ R ) + 1) ∪ n − [ i =1 ( O ∩ R ) + 1 + a i ) ( ⋆ ′ ) . As O is a valuation ring, 1 + a ∈ O or (1 + a ) − ∈ O . If it is the former then(( O ∩ R ) + 1) ∪ (( O ∩ R ) + 1 + a ) ⊆ O ∩ R and, as before, we get a contradiction to P-MINIMAL INTEGRAL DOMAINS 11 the index being n . If (1 + a ) − ∈ O then we get a contradiction after multiplying( ⋆ ′ ) by (1 + a ) − , as before. As a result, R = O ∩ R , i.e. R ⊆ O . (cid:3) Lemma 5.2.
Let R be a local domain and O a valuation ring of Frac( R ) . If O / ( R ∩ O ) is finite then R [ O ] (the ring generated by R and O ) enjoys the followingproperties(1) it is a valuation ring;(2) it is the integral closure of R ;(3) it is a finite extension of R and in particular, it is definable in the structure (Frac( R ) , R ) .Proof. If |O / ( R ∩ O ) | = 1 then R [ O ] = R is a valuation ring and there is nothing toshow, so we may assume that the index is greater than 1. Let R ′ = O ∩ R and let a , . . . , a n ∈ O be such that O = R ′ ∪ S ni =1 ( R ′ + a i ). Then O = R ′ [ a , . . . , a n ], and O is a finite R ′ -module, in particular R [ O ] = R [ a , . . . , a n ] is a finite R -module.As a result, R [ O ] is an integral extension of R and R [ O ] is definable in ( K, R ). As R [ O ] is a valuation ring (it contains the valuation ring O ) it is integrally closed andhence it is the integral closure of R . (cid:3) For valuation rings, the following lemma is a straight application of the approx-imation theorem for incomparable valuation rings. By assuming inp-minimality, italso follows for integrally closed rings.
Lemma 5.3.
Let R and R two integrally closed rings and K = Frac( R ) =Frac( R ) . If ( R , R ) is inp-minimal, then R ⊆ R or R ⊆ R .Proof. By inp-minimality, R / ( R ∩ R ) or R / ( R ∩ R ) is finite. Assume theformer, then R is an integral extension of R ∩ R . As R ∩ R is again integrallyclosed, we must have that R = R ∩ R , i.e. R ⊆ R . (cid:3) Definition 5.4.
Let ( R, M ) be a local ring and ( R ′ , M ′ ) a local overring of ( R, M ).We say that R ′ is a dominant extension of R if M = R ∩ M ′ (equivalently M ⊆ M ′ ), and non-dominant , otherwise. Proposition 5.5.
Let ( R, M ) be a dp-minimal domain K = Frac( R ) .(1) If O is a local non-dominant overring of R (not necessarily externally defin-able in ( K, R ) ), then O is a valuation ring and there exists a non-maximalprime ideal p of R such that O = R p .(2) If R ⊆ O is a local dominant valuation overring such that ( R, O ) is dp-minimal (e.g. if O is externally definable), then for any non-maximal primeideal p of R , O ⊆ R p . Furthermore, Spec( R ) \ { M } is an initial segmentof Spec( O ) \ { m } .Proof. (1) Since R ⊆ O is non-dominant, p = m ∩ R is a non-maximal prime ideal of R , where m is the maximal ideal of O . By Proposition 3.8, R p is valuation overringof R . As R p ⊆ O (if a ∈ R , s ∈ R \ p , then s / ∈ m hence a/s ∈ O ), we have that O is also a valuation ring. Since O is a localization of R p , O is a localization of R . (2) Let p be a non-maximal prime ideal of R . By Proposition 3.8, R p is a (non-dominant) valuation overring of R . Since ( R, O ) is dp-minimal, by Proposition 2.8,( R p , O ) is dp-minimal as well so by Lemma 5.3, O ⊆ R p or R p ⊆ O . As R ⊆ O isdominant, necessarily O ⊆ R p . We need now to show that p is a prime ideal of O .To show that it is an ideal, we note that p ⊆ p O ⊆ p R p = p , where the last equalityis by Theorem 3.9. The ideal p is a prime ideal of O since it is a prime ideal of R p .To show that it is an initial segment, we show that if q is a prime ideal of O strictlycontained in M then q is a prime ideal of R . Indeed, q ⊆ R and q R ⊆ q O ⊆ q so q is an ideal of R . Further it is prime in O so in particular in R . (cid:3) Corollary 5.6.
Let R be a domain and O a valuation ring of Frac( R ) such that ( R, O ) is dp-minimal. Then R ⊆ O or O ⊆ R .Proof. By dp-minimality, O / ( O ∩ R ) is finite or R/ ( O ∩ R ) is finite. If R/ ( O ∩ R ) isfinite, then by Lemma 5.1, R ⊆ O . Thus we may assume that O / ( O ∩ R ) is finite.If R/ ( O ∩ R ) is a non-dominant extension then, by Proposition 5.5 R is a valuationoverring of O ∩ R and hence by Lemma 5.3 R and O are comparable.So we may assume that R is a dominant overring of O ∩ R . We prove that R ⊆ O . Let a ∈ R . If a is non-invertible in R then a is non-invertible in everydominant valuation overring of R . By Lemma 5.2 R [ O ] /R is an integral extensionof local rings and hence a dominant extension, so a is non-invertible in R [ O ]. Itfollows that a − / ∈ O hence a ∈ O .Now, if a is invertible in R and a / ∈ O , then a − ∈ O ∩ R and a / ∈ O ∩ R , hence a − is in the maximal ideal of O ∩ R . Since R is a dominant extension of O ∩ R , a − is also in the maximal ideal of R , contradiction. (cid:3) Remark . In characteristic p >
0, Corollary 5.6 has a more direct proof. Let O be an externally definable valuation subring of K = Frac( R ). Lemma 5.1 takescare of the case where R/ ( O ∩ R ) is finite so we may assume that O / ( O ∩ R ) isfinite and set R ′ = O ∩ R . Since O is dp-minimal, by Theorem 4.1, F algp ⊆ O . As aresult, ( R ′ + F algp ) /R ′ ∼ = R ′ / ( F algp ∩ R ′ ) is finite as well. It is easy to see that everysubring of F algp is a field, hence F algp ∩ R ′ is a subfield of F algp of finite codimensionand in particular an infinite field. Hence R ′ contains an infinite field and so R ′ is a valuation ring by Theorem 4.1, and so is R . By Lemma 5.3, R and O arecomparable. 6. The Prime Spectrum of Dp-minimal Domains
We end with some non-elementary results concerning the prime spectrum ofcertain dp-minimal domains.Let R be a domain with Spec( R ) linearly ordered by inclusion (e.g. if R is inp-minimal). For p ( q ∈ Spec( R ), we will say that p is a predecessor of q and that q is a successor of p . For a ∈ M we observe the following: • p h a i is the minimal prime ideal containing h a i ; • P a is the maximal prime ideal not containing a ; • P a is the immediate predecessor of p h a i .Note that if a ∈ M then P a = M , if a = 0 then p h a i is not equal to the zeroideal and that P a for a = 0 does not make any sense. Lemma 6.1.
Let R be a domain with Spec( R ) linearly ordered by inclusion. For p ( q ∈ Spec ( R ) , the following are equivalent:(1) p is the immediate predecessor of q ;(2) p = P a and q = p h a i for any a ∈ q \ p .Proof. Assume that p is an immediate predecessor of q . For a ∈ q \ p , necessarily p = P a and q = p h a i by definition. The other direction follows by the abovediscussion. (cid:3) Proposition 6.2.
Let R be a domain with Spec( R ) linearly ordered by inclusion.The following are equivalent.(1) { P a | a ∈ M } \ { M } is densely ordered by inclusion;(2) there are no three consecutive elements in Spec( R ) ;(3) for all a, b ∈ M with a, b = 0 , P a = p h b i . P-MINIMAL INTEGRAL DOMAINS 13
Proof. (1) implies (2) is obvious. (2) implies (3) . Assume that P a = p h b i for some a, b ∈ M with a, b = 0. Bythe above discussion, P b ( p h b i = P a ( p h a i are three consecutive prime ideals,contradicting (2) . (3) implies (1) . Assume there are P a ( P b ( M , for a, b ∈ M , necessarily non-zero elements, with P a ( P b consecutive prime ideals. By Lemma 6.1, P b = p h c i and P a = P c for some c ∈ P b \ P a , contradicting the assumption. (cid:3) Definition 6.3.
We say that a domain R , with Spec( R ) linearly ordered by in-clusion, has property ( ⋆ ) if it satisfies one of the equivalent conditions of Proposi-tion 6.2.An integral domain of finite Krull dimension has property ( ⋆ ) if and only if it isone-dimensional. It is not hard to see that any ℵ -saturated domain has property( ⋆ ), and actually this is true in higher generality. Proposition 6.4.
Let D be a κ -saturated domain and R a W -definable local subringwhose prime ideals are linearly ordered. Then R has property ( ⋆ ) .Remark . By R W -definable we mean R is definable by W i<λ ϕ i ( x, a i ) for some λ < κ . Proof.
Assume that R is defined by W i<λ “ x ∈ R i ”, where λ < κ . There is no harmis assuming that the R i are closed under finite unions. Let M be the maximal idealand assume towards a contradiction that P a = p h b i , as prime ideals of R , for some a, b ∈ M with a, b = 0. Note that p h b i is W -definable: _ n ∈ N ( x n ∈ bR ) = _ n ∈ N (( ∃ y ∈ R )( x n = yb )) = _ n ∈ N _ i<λ (( ∃ y ∈ R i )( x n = yb )) . As for P a , it is defined by the type ^ n ∈ N ( a n / ∈ xR ) = ^ n ∈ N ( ¬ ( ∃ y ∈ R )( a n = xy )) = ^ n ∈ N ^ i<λ (( ∀ y ∈ R i )( a n = xy )) . Since P a = p h b i , by compactness there exist finite subsets F ⊆ N and I ⊆ λ suchthat x ∈ P a ⇐⇒ ^ n ∈ F ^ i ∈ I (( ∀ y ∈ R i )( a n = xy )) . Since the R i are closed by finite unions, there is some i < λ such that x ∈ P a ⇐⇒ ^ n ∈ F (( ∀ y ∈ R i )( a n = xy )) . Let n = max { n : n ∈ F } +1. Since a is non-invertible in R , a n y = a n for all y ∈ R (in particular for all y ∈ R i ) and n ∈ F . As a result, a n ∈ P a , contradiction. (cid:3) Remark . Let O be a valuation ring with value group Γ. It is not hard to see that O has property ( ⋆ ) if and only if the set of archimedean components of Γ is denselyordered. Indeed, by using the standard correspondence between prime ideals andconvex subgroups of the value group, each archimedean component corresponds to p h a i /P a for some 0 = a ∈ M . Remark . Let ∗ R be the hyperreals (resp. ∗ C the hypercomplex) and b R (resp. b C ) the ring of bounded elements. b R and b C are W -definable valuation rings in an ℵ -saturated domain, hence Proposition 6.4 applies. In [EK19], Echi and Khalfallahprove directly that these valuation rings do not have three consecutive prime ideals.The previous proposition is a generalisation of this result. Proposition 6.8.
Let ( R, M ) be a local ring and R ⊆ O a dominant valuationoverring with ( R, O ) dp-minimal. If O has property ( ⋆ ) then so does R .Proof. By Proposition 5.5 (2), if O does not have three consecutive prime idealsthen the same holds for Spec( R ). (cid:3) Corollary 6.9.
Let ( K, v ) be a dp-minimal field with a valuation ring O havingproperty ( ⋆ ) (e.g. if vK is ℵ -saturated). Let ( R, M ) be an externally definablering in ( K, v ) .(1) If v ( M ) > then R has property ( ⋆ ) .(2) If the residue field of ( K, v ) is finite then R has property ( ⋆ ) .Proof. (1) By Corollary 5.6, R ⊆ O or O ⊆ R . If the former holds, then applyProposition 6.8. If the latter holds, then R has property ( ⋆ ), since Spec( R ) is aninitial segment of Spec( O ). (2) As in (1) , we may assume that R ⊆ O . By Proposition 6.8, it is enough toshow that O is a dominant extension of R . Otherwise, by Proposition 5.5, O = R p for some non-maximal prime ideal p of R . As R is divided by Theorem 3.9, | R/ p | ≤ | R p / p | = | R p / p R p | < ∞ , contradicting Remark 3.7. (cid:3) In a domain with linearly ordered prime spectrum, ℵ -saturation also gives that P a = { } for all a . Divided domains satisfying this property are called pointwisenon-archimedean [Dob86b] [AKP98]. A somewhat mirror notion is requiring that p h a i 6 = M for all a , or equivalently, the maximal ideal does not have an immediateprime predecessor. Proposition 6.10.
Let R be a dp-minimal domain with maximal ideal M . If M has no immediate prime predecessor then(1) R is a pseudo-valuation domain;(2) M / h a i is infinite for all a ∈ M ;(3) M = M ;(4) M − = T a ∈ M R P a is a valuation overring with maximal ideal M ;(5) for every a, b ∈ M either a ∈ h b i or b ∈ h a i .Proof. (1) It is sufficient to prove that M is strongly prime. Assume that x, y ∈ Frac( R ) with xy ∈ M . As M does not have an immediate predecessor, p h xy i ( M (for otherwise, P xy is an immediate predecessor of M ). Since p h xy i is a primeideal, Proposition 3.17 gives x ∈ p h xy i or y ∈ p h xy i , as needed. (2) If M / h a i is finite for some a ∈ M then for every b ∈ M there exists some n and m such that b n − b m ∈ h a i hence b k ∈ h a i for some k , i.e. M = p h a i .Contradicting the fact that M does not have an immediate predecessor. (3) Assume there exists a ∈ M \ M . Since M does not have an immediatepredecessor p h a i ( M and as p h a i is a prime ideal and M is an ideal, byRemark 3.2 and Theorem 3.9, p h a i and M are comparable. Since a ∈ p h a i but a / ∈ M , necessarily M ⊆ h a i ⊆ p h a i , but it follows from (2) that M / h a i isunbounded, and hence so is M / M , contradiction. (4) Consider ˇ R = T a ∈M R P a . Since the prime ideals are linearly ordered, byProposition 3.8, ˇ R is a valuation overring of R . Its maximal ideal is S a ∈M P a R P a = S a ∈M P a , where the last equality holds since R is divided by Theorem 3.9. Since M has no prime predecessor, by Lemma 3.3 we may conclude that S a ∈M P a = M .The result now follows by [HH78, Theorem 2.7 and Theorem 2.10]. (5) follows from (4) and [Bad95, page 4368]. (cid:3) P-MINIMAL INTEGRAL DOMAINS 15
Remark . If R has an immediate prime predecessor, M = p h a i for some a ∈ M . By Theorem 3.9, there exists an n ∈ N such that b n ∈ h a i for all b ∈ M .This fits well with the examples at the end of Section 4. For each of these kind ofexamples, R is is a pseudo-valuation ring or there exists some ideal h a i and n ∈ N such that b n ∈ h a i for all b ∈ M . Remark . A domain R is said to be fragmented if for each non-invertible a ∈ R there exists a non-invertible b ∈ R such that a ∈ T n h b n i [Dob86a, CD01]. Ina divided domain, P a = T n h a n i , and it is easy to see that a divided domain isfragmented if and only if the maximal ideal has no immediate prime predecessor.Consequently, every fragmented dp-minimal domain is a pseudo-valuation ring.If O is a dp-minimal domain of positive characteristic having property ( ⋆ ) thenfor any a ∈ O , F p + P a is a fragmented pseudo-valuation domain which is not avaluation ring. References [Aki67] Tomoharu Akiba. A note on AV-domains.
Bull. Kyoto Univ. Educ., Ser. B , 31:1–3,1967.[AKP98] D.D. Anderson, B.G. Kang, and M H. Park. Anti-archimedean rings and power seriesrings.
Communications in Algebra , 26(10):3223–3238, 1998.[Bad95] Ayman Badawi. On domains which have prime ideals that are linearly ordered.
Com-munications in Algebra , 23(12):4365–4373, 1995.[BS77] Monte B. Boisen, Jr. and Philip B. Sheldon. CPI-extensions: overrings of integraldomains with special prime spectrums.
Canadian J. Math. , 29(4):722–737, 1977.[CD01] Jim Coykendall and David Dobbs. Fragmented domains have infinite krull dimension.
Rendiconti del Circolo Matematico di Palermo , 50(2):377–388, 2001.[CR76] Gregory L. Cherlin and Joachim Reineke. Categoricity and stability of commutativerings.
Ann. Math. Logic , 9(4):367–399, 1976.[CS80] Gregory Cherlin and Saharon Shelah. Superstable fields and groups.
Ann. Math. Logic ,18(3):227–270, 1980.[DDAa00] Sarah Glaz (eds.) D. D. Anderson (auth.), Scott T. Chapman.
Non-Noetherian Com-mutative Ring Theory . Mathematics and Its Applications 520. Springer US, 1 edition,2000.[DGL11] Alfred Dolich, John Goodrick, and David Lippel. Dp-minimality: basic facts and ex-amples.
Notre Dame J. Form. Log. , 52(3):267–288, 2011.[Dob76] David E. Dobbs. Divided rings and going-down.
Pacific J. Math. , 67(2):353–363, 1976.[Dob78] David E. Dobbs. Coherence, ascent of going-down, and pseudo-valuation domains.
Houston J. Math , pages 551–567, 1978.[Dob86a] David E. Dobbs. Fragmented integral domains.
Portugaliae mathematica , 43(4):463–473, 1985-1986.[Dob86b] David E. Dobbs. Ahmes expansions of formal laurent series and a class of nonar-chimedean integral domains.
Journal of Algebra , 103(1):193 – 201, 1986.[DW20] Jan Dobrowolski and Frank O. Wagner. On ω -categorical groups and rings of finiteburden. Israel J. Math. , 236(2):801–839, 2020.[EK19] Othman Echi and Adel Khalfallah. On the prime spectrum of the ring of boundednonstandard complex numbers.
Proc. Amer. Math. Soc. , 147:687–699, 2019.[HH78] John R. Hedstrom and Evan G. Houston. Pseudo-valuation domains.
Pacific J. Math. ,75(1):137–147, 1978.[HHJ19] Yatir Halevi, Assaf Hasson, and Franziska Jahnke. Definable v-topologies, henselianityand NIP. preprint, https://arxiv.org/abs/1901.05920 , 2019.[HP18] Nadja Hempel and Daniel Palac´ın. Division rings with ranks.
Proc. Amer. Math. Soc. ,146(2):803–817, 2018.[HP19] Yatir Halevi and Daniel Palac´ın. The Dp-rank of abelian groups.
J. Symb. Log. ,84(3):957–986, 2019.[Joh16] Will Johnson.
Fun with Fields . PhD thesis, University of California, Berkeley, 2016.[Joh18] Will Johnson. The canonical topology on dp-minimal fields.
J. Math. Log. ,18(2):1850007, 23, 2018.[Joh19] Will Johnson. Dp-finite fields i: infinitesimals and positive characteristics. preprint, https://arxiv.org/abs/1903.11322 , 2019. [Joh20] Will Johnson. Dp-finite fields vi: the dp-finite shelah conjecture. preprint, https://arxiv.org/abs/2005.13989 , 2020.[JSW17] Franziska Jahnke, Pierre Simon, and Erik Walsberg. Dp-minimal valued fields.
J. Symb.Log. , 82(1):151–165, 2017.[KS13] Itay Kaplan and Saharon Shelah. Chain conditions in dependent groups.
Ann. PureAppl. Logic , 164(12):1322–1337, 2013.[KSW11] Itay Kaplan, Thomas Scanlon, and Frank O. Wagner. Artin-Schreier extensions in NIPand simple fields.
Israel J. Math. , 185:141–153, 2011.[Mac71] Angus Macintyre. On ω -categorical theories of fields. Fundamenta Mathematicae ,71(1):1–25, 1971.[Mil19] C´edric Milliet. NIP, and NTP2 division rings of prime characteristic. preprint, https://arxiv.org/abc/1903.00442 , 2019.[Oka84] Akira Okabe. Some results on pseudo-valuation domains.
Tsukuba J. Math. , 8(2):333–338, 12 1984.[OPP96] Margarita Otero, Ya’acov Peterzil, and Anand Pillay. On groups and rings definable ino-minimal expansions of real closed fields.
Bull. London Math. Soc. , 28(1):7–14, 1996.[Pic76] Gabriel Picavet. Id´eaux premiers de Goldman. In
Deuxi`eme Colloque d’Alg`ebre Com-mutative (Rennes, 1976), Exp. No. 7 , page 7. D´epartement de Math´ematiques etInformatique, Universit´e de Rennes, 1976.[PS99] Ya’acov Peterzil and Charles Steinhorn. Definable compactness and definable sub-groups of o-minimal groups.
J. London Math. Soc. (2) , 59(3):769–786, 1999.[Sim15] Pierre Simon.
A guide to NIP theories , volume 44 of
Lecture Notes in Logic . Associationfor Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge, 2015.[TZ12] Katrin Tent and Martin Ziegler.
A course in model theory , volume 40 of
Lecture Notesin Logic . Association for Symbolic Logic, La Jolla, CA; Cambridge University Press,Cambridge, 2012. † Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram9190401, Jerusalem, Israel
E-mail address : [email protected] URL : http://choum.net/~chris/page perso/ ∗ Department of mathematics, Ben Gurion University of the Negev, Be’er Sehva,Israel
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