ON PSEUDO-VALUATION DOMAINS WHICH ARE NOT VALUATION DOMAINS

In this article, we discuss the n-root closedness, root closedness, seminormality, S-root closedness, S-closedness, F -closedess of PVDs. A valuation domain, being integrally closed, is obviously root closed. So our interest of study is for a class of non-valuation PVDs. Let R ⊂ B be a domain extension such that R is a PVD and the common ideal P of R and B is a prime ideal in R. If R is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F -closed) in B, then R/P is PVD, which is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F -closed) in B/P . Further we study the relationship of atomic PVDs to atomic PVDs, SHFDs, LHFDs and BVDs. We also discuss a relative ascent and descent in general and particularly for the antimatter property of PVDs. Mathematics Subject Classification 2010: 13A15, 13A18, 13B30, 13F30


Introduction
Following [3], an integral domain R with quotient field K is said to be root closed if whenever x n ∈ R for some x ∈ K, and n ∈ Z + , then x ∈ R. We define R to be n-root closed if whenever x n ∈ R for some x ∈ K, then x ∈ R. Any integral domain R is trivially 1-root closed.An integral domain R is root closed if and only if it is n-root closed for each n ∈ Z + .Further, R is seminormal if whenever x ∈ K with x 2 , x 3 ∈ R, then x ∈ R.More generally, if R ⊆ B be a unitary commutative ring extension, then R is n-root closed in B if whenever x ∈ B with x n ∈ R, then x ∈ R. By [19], R is F -closed in B if whenever x ∈ B with x 2 , x 3 ∈ R and nx ∈ R for some n ∈ Z + , then x ∈ R. Following [8, Page 2], A is S-root closed in B if whenever b is in B and b n is in A for some n in S, then b is in A. The ring A is called S-closed in B if b is in B and b n is in A for all n in S, then b is in A.
According to Cohn [12], an integral domain R is an atomic domain if each nonzero non-unit of R is a product of a finite number of irreducible elements (atoms) of R.

TARIQ SHAH AND WAHEED AHMAD KHAN
Acccording to [21], an atomic domain R is a half-factorial domain (HF D) if for each nonzero nonunit element x ∈ R, if x = x 1 ...x m = y 1 ...y n with each x i , y j , 1 ≤ i ≤ m, 1 ≤ j ≤ n is irreducible element in R, then m = n.
According to [15], a prime ideal P of an integral domain R with quotient field K, is known as strongly prime if x, y ∈ K such that xy ∈ P , then either x ∈ P or y ∈ P .An integral domain R is said to be pseudo-valuation domain (P V D) if each prime ideal of R is strongly prime.
This study includes an investigation of "When is a non-valuation PVD n-root closed (resp., root closed)?"Note that a valuation domain is a PVD (cf.[15,Proposition 1.1]).But a valuation domain is integrally closed and hence root closed, so our interest is in a P V D which is not a valuation domain.In the first part of this study we establish some conditions under which a P V D (which is not a valuation domain) is n-root closed (respectively root closed).We apply these results to obtain seminormal, S-root closed and S-closed P V Ds.We also discuss the behavior of root closure of R/P, where P is a prime ideal of R, whenever R is root closed P V D. In [8], S-root closure in factor ring of a root closed ring is discussed (cf.[8, Theorem 1.8]), we extend it for n-root closure and for root closure.We also generalize [8,Proposition 1.5] for n-root closure and for root closure for extension of P V Ds.
According to [4], an HF D is strongly half-factorial domain (SHF D) if each of its overrings is an HF D. An HF D is locally half-factorial domain (LHF D) if each of its localization is an HF D. Following [16], let R be an HF D with quotient field K.If R = K, we define the boundary map δ R : K * → Z by δ R (α) = t − s, where α = (x 1 ...x t )/(y 1 ...y s ) ∈ K and x i , y j are irreducible elements in R.
By [16], an integral domain R with quotient field K, is called boundary valuation domain (BV D) if R is an HF D, and for any α ∈ K with δ R (α) = 0, either α ∈ R or α −1 ∈ R, where δ R is boundary map defined on K.
We also discuss the relationship of atomic P V Ds in which we relate atomic P V Ds to SHF Ds, LHF Ds and BV Ds.Further we discuss a relative ascent and descent of P V Ds.
Finally we emphasize upon the antimatter property (a domain R is an antimatter domain if it has no atoms [13]) of pseudo-valuation domain while considering the condition * .Here we generalized a few of results of [13] relative to the condition * .

Root closure in PVDs
We begin with the following.
Remark 2.1.Let V be a valuation domain of the form K + M, where K is any field and M is the maximal ideal of V .If F is a proper subfield of K which is Let S be a multiplicative submonoid of P, generated by some set of positive primes.Here, an increasing sequence of subfields of R can be defined by [6, p-7]).According to [20], an integral domain R, with quotient field K, is said to be pre-Schreier domain if for all x, y, z ∈ R\{0}, x | yz implies x = rs, where r, s ∈ R with r | y and s | z.An integrally closed pre-Schreier domain is called a Schreier domain.
Remark 2.3.In Remark 2.2, the maximal ideal (P ) T is idempotent, therefore is an example of a pre-Schreier PVD which is not a Schreier domain but is root closed.
The above observations yield the following: Lemma 2.4.Let V be a valuation domain of the form K + M, where K is a field and M be the maximal ideal of V and F be a proper subfield of K. (1) let L be the algebraic closure of Q and let F be the subfield of L consisting of all elements α over Q such that the minimal polynomial for α over Q is solvable.Choose β ∈ L but not in F , where K is a field, contains n th root of unity, not in R.
In [8], S-root closure of commutative ring extensions R ⊆ B ⊆ C has already been discussed (see [8,Proposition 1.5]).We are looking at it for n-root closure and root closure particularly for the extensions of P V Ds.
Corollary 2.7.Let R ⊆ B ⊆ C be extensions of P V Ds. ( ( ( So R is seminormal in C by Proposition 2.6.Conversely, let R be seminormal in by Proposition 2.6.Hence R is seminormal in B. (3) Let R be F -closed in B. That is for any x ∈ B whenever nx, x 2 , x 3 ∈ A, In [8], it is established that, for a commutative ring extension R ⊆ B, the factor ring R/I is S-root closed, where R is a root closed ring and I is a common ideal of R and B (see [8,Theorem 1.8]).We focus on this situation for prime ideal P of R which is also an ideal in B, instead of I and discussed the root closure of factor ring R/P, whenever R is root closed P V D. Furthermore we also address the seminormality and F -closedness for factor ring R/P of a P V D R.
In Theorem 2.8, we prove the result specially for pseudo-valuation domains.
Theorem 2.8.Let B be a domain extension of a P V D R such that P is a prime ideal of R which is also an ideal in B. If R is n-root closed in B, then R/P is P V D, which is n-root closed in B/P.
Proof.By [10, Corollary 3], R/P is P V D. Let (x + P ) n ∈ R/P , where x + P ∈ B/P.This implies x n + P ∈ R/P, where Corollary 2.9.Let B be a domain extension of a P V D R such that P is a prime ideal of R which is also an ideal in B. Then with x 2 , x 3 ∈ R and nx ∈ R for some positive integer n, then x ∈ R. Let (x + P ) 2 , (x + P ) 3 ∈ R/P.This means x 2 + P, x 3 + P ∈ R/P.Let n(x + P ) ∈ R/P, then nx + P ∈ R/P, where x + P ∈ B/P and so nx ∈ R. As R is F -root closed, so x ∈ R, which implies that x + P ∈ R/P .Hence R/P is F -closed in B/P.

Atomic PVDs
In this part we consider the case of atomic P V Ds and relate it with SHF Ds, LHF Ds and BV Ds.
Recall from [4] that, an HF D is SHF D if each of its overring is an HF D. is an atomic P V D which is not an F F D.
Following [16], let R be an HF D with quotient field K.If R = K, we define the boundary map δ R : K * → Z by δ R (α) = t − s, where α = (x 1 ...x t )/(y 1 ...y s ) ∈ K and x i , y j are irreducible elements in R.
Recall from [16] that, an integral domain R with quotient field K, is called BV D if R is an HF D and for any α ∈ K with δ R (α) = 0 either α ∈ R or α −1 ∈ R, where δ R is boundary map defined on K.
Theorem 3.4.The following assertions are equivalent for an integral domain R.
(1) R is an atomic P V D.
(2) R is a BV D. Proof.

A relative ascent and descent
We recall the following as in [17].(b) The domain extension   Proof.Let R be an antimatter domain and T be its overring such that u ∈ U (T ) ⊆ F and each t ∈ T can be written as t = ur by Condition * .Then by Proposition 4.8, it is clear that if R is an antimatter domain then T is an antimatter domain.
The following lemma is the converse of [13,Cor 3.3(b)].

Remark 2 . 2 .
The additive monoid monoid S = Q + ∪ {0} form algebra R[X; S] over R, which is Bezeout domain (cf.[20, Example 4.5]).Let P = {f ∈ R[X; S] : f has zero constant term}.P is a prime ideal and clearly the multiplicative system T = R[X; S]\P is the set of elements with nonzero constant terms.It is easy to see that R[X; S] = R + P and as it is Bezeout domain, so (R[X; S]) T = R + (P ) T = V is a valuation domain with maximal ideal (P ) T (cf.[20, Example 4.5]).Then R = K S + (P ) T , where K S = ∪K n .Thus R = K S + (P ) T is n-root closed, as K S is n-root closed in R, by [6, Lemma 3.2].So, R is n-root closed P V D which is not a valuation domain.

Proposition 3 . 1 .Remark 3 . 2 . 1 .
A Noetherian P V D is an SHF D. Proof.Each overring of R is P V D (cf.[15, Corollary 3.3]).Since R is Noetherian, therefore its integral closure R is Noetherian and dim(R ) is 1 (cf.[15, Corollary 3.4]).Obviously R is a Dedekind domain.By [14, Theorem 1], every overring of R is atomic.That is each overring of R is an atomic P V D. So each overring of R is an HF D by [9, Theorem 5].Hence R is an SHF D. Following [4], an HF D is LHF D if each of its localization is an HF D. By definitions of SHF D and LHF D it is clear that an SHF D is an LHF D. Therefore a Noetherian P V D is also an LHF D because it is an SHF D by Proposition 3.Remark 3.3.(i)Let (R, M ) be a P V D which is not a valuation domain andM 2 = M ,then R is a pre-Schreier [20, Theorem 4.4].(ii) An atomic P V D may not be a finite factorization domain (F F D), (By [1], R is a finite factorization domain (F F D) if each nonzero non-unit element of R has a finite number of non-associate divisors and hence, only a finite number of factorizations upto order and associates), because R + XC[[X]]

Condition * :
Let R ⊆ B be a unitary (commutative) ring extension such that U (B) represent the set of units of B. For each b ∈ B there exist u ∈ U (B) and a ∈ R such that b = ua.For a unitary (commutative) ring extension R ⊆ B, the conductor of A in B is the largest common ideal R : B = {a ∈ R : aB ⊆ R} of R and B. The following are few examples of unitary (commutative) ring extensions which are satisfying Condition * .Remark 4.1.[17, Example 1] (i) If B is a field, then the ring extension R ⊆ B satisfies Condition * .(ii) If B is a fraction ring of R, then the ring extension R ⊆ B satisfies Condition * .Hence the ring extension R ⊆ B satisfies Condition * is the generalization of localization.(iii) If the ring extensions R ⊆ B and B ⊆ C satisfy Condition * , then so does the ring extension R ⊆ C. (iv) If the ring extension R ⊆ B satisfies Condition * , then the extensions of rings R + XB[X] ⊆ B[X] and R + XB[[X]] ⊆ B[[X]] satisfy Condition * .There are number of examples of domain extensions R ⊆ B which are satisfying Condition * but the conductor ideal R : B is not a maximal ideal of R, as the following remark shows.Remark 4.2.(i) Following [2, Example 5.3], let V be a valuation domain such that its quotient field K is the countable union of an increasing family {V i } i∈I of valuation overrings of V. Let L be a proper field extension of K thus, L * /K * is infinite as indicated in [2, Example 5.3].(a) The domain extension R = V i + XL[[X]] ⊆ L[[X]] = B satisfies the Condition * as the extension V i ⊆ L satisfies the Condition * .But XL[[X]] is not a maximal ideal in R and such that U (R) = U (B).

4. 1 .
The case of PVDs.In the following we observe that the ascent of P V D holds for a domain extension R ⊆ B which satisfies Condition * .Theorem 4.3.Let R ⊆ B be a domain extension which satisfies Condition * .If R is a P V D, then B is a P V D. Proof.As R is a P V D, [9, Proposition 4] says: A ring R is a PVR iff for all a, b ∈ R either a | b or every proper divisor of b divides a.Since R is a P V D, therefore either a | a 1 or any divisor d of a 1 divides a.Let a | a 1 clearly b 1 | b 2 .Now suppose b 1 does not divide b 2 and let d | b 2 which implies

Corollary 4 . 4 .
Let R ⊆ B be a domain extension which satisfies Condition * and M = R : B be a maximal ideal of R .If R is an atomic P V D, then B is an atomic P V D. Proof.Let R be an atomic P V D. By [17, Proposition 2.6 (a)], B is atomic.By Theorem 4.3, B is P V D. Hence B is an atomic P V D. The following proposition provides the sufficient condition for [13, Proposition 3.2(b)].Proposition 4.8.Let R ⊆ B be the domain extension such that Spec(R) = Spec(B), which satisfies Condition * , then B is an antimatter domain if and only if R is an antimatter domain.Proof.Assume that R is an antimatter domain, then by Condition * for r ∈ R there exists t ∈ B such that t = ur, with u ∈ U (B) and r ∈ R.This means t and r are assosiates in divisibility in B and by Lemma 4.7, t is not an atom in B. Thus B is an antimatter domain.The converse follows by [13, Proposition 3.2(b)].

Lemma 4 . 9 .
Let R be an integral domain with quotient field F satisfying Condition * .If R is an antimatter domain, then any overring of R having same spectrum to R is an antimatter domain.

Lemma 4 . 10 .
Let R be a PVD and the extension R ⊆ V satisfies Condition * , where V is the canonically associated valuation overring of R .If R is an antimatter domain, then V is an antimatter domain.Proof.The result is obvious by the irreducibility of an elements in R and its overring V , by adding Condition * gives the result that V is an antimatter domain if R is an antimatter domain.Example 4.11.The domain extensionR = V i + XL[[X]] ⊆ L[[X]] = B satisfies the Condition * as the extension V i ⊆ L satisfies the Condition * .But XL[[X]]is not a maximal ideal in R and such that U (R) = U (B).Then clearly if R is antimatter domain then so is B, as irreducibility in R implies irreducibility in B.

Example 4 . 12 .
Let V = F + M be a nontrivial valuation domain, where F is a field and M the maximal ideal of V .Let D be a domain with quotient field F satisfying Condition * , and put R = D + M .Indeed R is an antimatter domain if and only if D is antimatter domain by [13, Corollary 3.10] with Condition * .