# Questions Posted

1. QUESTION (HD0301): If R=F[X³,X^{4}],I=(X,X,X) then how is I¹=F[X]? (Here F is a field)
2. QUESTIONS (HD0302): How is any prime ideal minimal over a t-ideal a prime t-ideal? How can you show that a maximal t-ideal is prime? How is a maximal height-one prime ideal a prime t-ideal?
3. QUESTION (HD0303): How do you show that a one-dimensional Bezout domain is completely integrally closed?
4. QUESTION (HD0304): If (V,M) is a valuation domain and X an indeterminate over V then how is V[X]_{M[X]} a valuation domain?
5. QUESTION (HD0305): If D is a PVMD and X an indeterminate over D then how can we show that D[X]is a PVMD?
6. QUESTION (HD0306): If D is a Prufer v-multiplication domain and if Q is a prime t-ideal of D then how is Q[X] a prime t-ideal of D[X]?
7. QUESTION (HD0307): Let S={X^{α}:α \in Q} where Q denotes the set of nonnegative rational numbers. Let R be the semi-group ring Q[S]. If I=(X-1)R is a radical ideal?
8. QUESTION (HD0308): Let S={X^{α}:α \in Q} where Q denotes the set of nonnegative rational numbers. Let R be the semi-group ring Q[S] and if P is a nonzero prime ideal of R, must P¹=R?
9. QUESTION (HD0309): If D is an integral domain, and M a prime ideal of D[X] with MD=(0) then how is D[X]_{M} a valuation domain?
10. QUESTION (HD0310): If D is completely integrally closed then how is D integrally closed?
11. QUESTION (HD0311): Is a prime t-ideal P, of a domain R, always a maximal t-ideal? Give an example if the answer is no.
12. QUESTION (HD0312): If P is a prime t-ideal of an integral domain R, must PR_{P} be a prime t-ideal of R_{P}?
13. QUESTION: (HD0313): Is every essential domain a P-domain?
14. QUESTION:(HD0314): What is a pullback? Give some examples.
15. QUESTION (HD0401): If S is a saturated multiplicative set in a domain R and if a,b are two nonzero elements of R such that (a/b) belongs to R_{S}, must b be in S?
16. QUESTION (HD0402): Is there an example of a prime t-ideal P of R that R_{P} is not a valuation domain?
17. QUESTION (HD0403): If R is a Prufer v-multiplication domain such that every maximal t-ideal of R is a maximal ideal, must R be a Prufer domain?
18. QUESTION:(HD0404) I wonder if there is a way to describe the (fractional) overrings of D+XK[X]. In particular, how can one find the (fractional) overrings of Z+XQ[X]? Would you be willing to suggest to me any papers or references to help me answer the above question?
19. QUESTION:(HD0405) Let there be a family {P_{α};α \in I} of prime ideals of R such that:
(1) Each R_{P_{α}} is a valuation domain and P_{α}R_{P_{α}} is divisorial
(2) the family {R_{P_{α}}:α \in I} is a family of finite character for R
(3) each pair of {R_{P_{α}}:α \in I} are independent.
Why for each maximal t-ideal,M, of R there is α \in I such that M=P_{α}?
20. QUESTION:(HD0501) In Huneke's book "Tight Closure and Its Applications", he mentioned the following fact regarding complete integral closure (pg. 14, Example 1.6.1): Let R be a Noetherian integral domain with fraction field K. Let α be an element in K. If there is a nonzero c \in R such that c(αⁿ) \in R for infinitely many n, then α is integral over R. I couldn't figure out how to show this, although I understand why this is true when "infinitely many" is replaced by "all", which is the definition of almost integral.
21. QUESTION:(HD0502). I have a problem with determining the properties of the ring R=Z[(1+√(-19))/2]. I suppose that it is a UFD and it is not a Euclidean domain. Also, I supose that it is a PID. What could you tell me about it?
22. QUESTION (HD 0503): Let a, b be integers such that b^{r}| a^{s} where r, s are natural numbers such that r ≥ s. Show that b| a.
23. QUESTION (HD 0504): Is it true that if I is a *-ideal of an integral domain D, for some star operation *, then the radical √I is also a *-ideal?
24. QUESTION (HD0601): Call an integral domain D an irreducible divisor finite (idf) domain if every nonzero element of D is divisible by at most a finite number of non-associated irreducible elements. Let K be a field, let Q^{+} be the set of non-negative rationals and let R=K[X;Q^{+}] be the monoid ring construction. Is there a field K such that K[X;Q^{+}] is not an idf domain?
25. QUESTION (HD0602): Kaplansky, in his book on Commutative Rings, calls an integral domain D an S-domain if for every height one prime P of D we have height(P[X])=1 where X is an indeterminate over D. He then moves on to define strong S-rings, to show that if R is a Noetherian ring then dim(R[X]) = dim(R)+1. Are there any examples of S-domains, or were they introduced to flash Seidenberg's name?
26. QUESTION (HD0701). Consider the following argument. Let R be a pre-Schreier domain. Then S = R\{0} is a saturated multiplicative set of completely primal elements. Now R_S [X] = (R[X])_S is a GCD domain and hence a Schreier domain. So by your version of Cohn’s Nagata type theorem R[X] is a pre-Schreier domain [Manuscripta Math. 80(1993), Corollary 8]). But according to MacAdam and Rush’s work R[X] pre-Schreier implies R[X] Schreier. What is the reason for this discrepancy?
27. QUESTION: (HD0702) Kronecker had associated, via "Kronecker function rings", a UFD with each ring of algebraic numbers years before Dedekind proved unique factorization of ideals of a ring of algebraic integers, of a special kind. Then why is it that we see Dedekind and Dedekind domains everywhere yet no mention of Kronecker? This is a, sort of, preliminary response. If any readers have an idea of how this question should be answered, they are welcome to write to me at zafrullah@lohar.com
28. QUESTION: (HD0703). Is it correct to define a Prüfer domain as a domain whose finitely generated ideals are invertible?
29. QUESTION: (HD0704). Is there a reference work on v-domains?  ANSWER: Click at the highlighted part: http://www.lohar.com/researchpdf/QA_session_on_v_domains.pdf
30. QUESTION: (HD0801) (Asked in person long ago.) Is there a good, brief, introduction to ideal systems in monoids from Ring-theoretic point of view?  ANSWER: Check out the answer to HD0704. I have provided some introduction which Professor Halter-Koch approved saying: “I have studied the new version of your question/answer session on v-domains. I was delighted to see that (for the first time?) the theory of ideal systems is mentioned in an adequate way.”
31. QUESTION: (HD0802) I am interested in learning about the generalizations of Prüfer domains called v-domains and Prüfer v-multiplication domains, but they are studied using the star operations, which I am not very familiar with. Is there a way of defining these concepts without any mention of star operations?
32. QUESTION: (HD0803). In your article, "Putting t-invertibility to use", you mention on page 443 that you have an example of a t-linked extension that is not t-compatible and doesn't satisfy any of (a)-(d) on pages 442 and 443, and that this example would be included in another article. I was wondering if you could let me know some such examples or could point me to a reference. (This question was asked by Jesse Elliott of CSU Channel Island.)
33. QUESTION: (HD0805) Let R be an integral domain which satisfies ACCP and I a non zero ideal of R. If R/I is an integral domain which is a homomorphic im.age of R Does R/I also satisfy ACCP?
34. QUESTION: (HD0806)  I was studying your paper "Factorization of certain sets of polynomials in an integral domain". In Theorem 5 of the paper for the proof of (1) <--> (2) you are quoting the reference of Arnold and Gilmer's paper, "on the contents of polynomials". But this paper does not contain the proof of the Result: Let D is an integral domain with identity having quotient field K. Then (1) If D is a Schreier ring, then for any positive integer n, D [X1…Xn] is inert in K [X1…Xn]. (2) If D [X1…Xn] is inert in K [X1…Xn] for some n greater than or equal to 1, then D is a Schreier ring. For that the author is writing that referee has communicated to the author. Sir do you have the proof of above result. If you have, then please send me.
35. QUESTION: (HD0807) In Cohn's paper "Bezout rings and their subrings" theorem 2.4 has no proof, and I was able neither to find any source of it, nor build it by myself. You may refer me to other material or internet.
36. QUESTION: (HD0901) In multiplicative ideal theory , we often deal with Picard groups. I want to know completely about the Picard groups. I also see in some materials that it has connections with Algebraic geometry. Please guide me to know about the Picard groups deeply. Which readings you suggest to be most useful?
37. QUESTION: (HD0902) How do you construct integrally closed domains that are not PVMD's? What's the simplest such example known?
38. QUESTION: (HD 0903) How do you construct a PVMD that is not Krull nor Prufer nor GCD? What's the simplest such example known?
39. QUESTION: (HD 0904) Let A \subseteq B be an extension of integral domains, let X be an indeterminate over B and let R= A + XB[X]. Under what conditions is X (a) an irreducible element of R (b) a prime element of R?
40. QUESTION: (HD1001) You refer to Conrad's F-condition a lot, in lattice ordered groups G, their generalizations, and in the so called multiplicative ideal theory; and it confuses me. I keep worrying about a situation in an l.o. group G where the condition F holds yet for some 0< a G we have that for every n N there is a set E_{n} consisting of pairwise disjoint elements below a. I would like to see a direct proof or an explicit reference where it is shown that the above situation cannot occur.
41. QUESTION: (HD1002)Everyone tells me that an integral domain that satisfies ACCP is atomic but no one shows me how. Could you please?  Find answer at: http://www.lohar.com/mithelpdesk/hd1002.pdf
42. QUESTION: (HD1101) I have the following question. It is taken from the exercises in Kaplansky’s book. Let R be a Prufer domain. Let P be a finitely generated prime ideal. Prove that P is maximal. Before the set of exercises, only 3 things have been proved. 1) Definition of Prufer domain, i.e every finitely generated ideal is invertible. 2) Invertible implies locally principal. 3) Localization of a prufer domain at a prime or maximal ideal is a valuation domain. Using these 3 facts, how can one give a proof of the above exercise.

44.

45.  QUESTION: (HD1103) Let R be a commutative ring. Can we say anything nice about R if we know that the set of zero divisors of R is a prime ideal? (This interesting question was asked by Viji Thomas from TIFR, Mumbai, India.)

46.  QUESTION: (HD1104) In HD1103 you have used two terms: primal ideal and primal element. Are they related? Can I say that the principal ideal generated by a primal element is a primal ideal? (I recall that some authors call an element x of a domain R primary if xR is a primary ideal.)

47.  QUESTION: (HD1105) Must an almost factorial domain be locally factorial?

53.  QUESTION: (HD 1207)

54.  QUESTION: (HD 1208)   Let R be a Noetherian local domain and let P be a height one prime ideal of R. Can we find an element x in P such that P is the only minimal prime containing x.

55.  QUESTION: (HD 1209) Why is the D+XD_S[X] construction from a GCD domain a Schreier domain? Could you give an example of a D+XD_S[X] construction from a GCD domain D such that D+XD_S[X] is not GCD?

56.  QUESTION: (HD 1210) What should we expect from an ideal A of grade 1 with A_{t}=D?

57.  QUESTION: (HD 1301) Anh, Marki and Vamos in [Trans. Amer. Math. Soc. 364 (8)(2012), 3967-3992 ] seem to suggest that what they call Bezout monoids provide the best set up for studying GCD domains and UFDs. Any comments?

58.  QUESTION: (HD 1302) Let L/K be a fields extension such that L is algebraic over K. Does there exists a positive integer m such that every irreducible element of K[X] (polynomial ring over K) has factorization of finite length less than m in L[X] ? If the response is "no", what are the couple (K,L) on which the response is "yes".

61.  QUESTION: (HD 1403) Anderson has shown that a locally principal ideal is a t-ideal. Can't we use the same argument to show that every w-locally principal ideal is a t-ideal?

63.  QUESTION: (HD 1405) Why  is $R:=Z+Q[X]$ an H-domain?

65.  QUESTION: (HD 1501) Let R be an integral domain and J be an ideal of R which is not contained in any non principal prime ideal of R. Is this ideal J principal? (Umar Nazir of Department of Mathematics, COMCASTS, Attok, Pakistan, asked this question.)

66.  QUESTION: (HD 1502) Is there an example of integral domain R, with fraction field K, such that, for some maximal ideal P of R, there exists a place of K extending the natural surjection from R to R/P, whose value field is a non trivial algebraic extension of R/P? This question was proposed by Michaël Bensimhoun, who also contributed with interesting remarks and examples.

67.  QUESTION: (HD1503) Let AB be an extension of integral domains such that for all divisorial ideals I of A we have: (c_{v}): I¹B = (IB)¹. Is it equivalent to the condition (c) of your article [ABZ, J. Algebra Appl. 11 (2012), no. 1, 1250007, 18 pp]? if not, is the extension t-linked? (we can suppose that both A and B are Krull domains but neither A nor B is Dedekind, because if A is Dedekind or reflexif (c_{v}) = (c)) ((c): I¹B = (IB)¹ for all I F(A)). (Walid Maaref, a Tunisian student, asked this question.)

70.  QUESTION (HD 1506): Please give an example to show that an Almost-Schreier domain is not a pre-Schreier domain generally.

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77.      QUESTION (HD 1802): In a personal communication, Professor Gyu Whan Chang wrote: I have the following objection to the proof of your Theorem 1 of your paper, with Tiberiu Dumitrescu, on, "Characterizing domains of finite *-character" (JPAA 214 (11(2010) 2087-2091.) In line -4 ~-1 of page 2088,  you said that "If V_n is not homogeneous, then V_n is contained in at least two *-comaximal elements which are *-comaximal with V_1, ..., V_{n-1}. This contradicts  the maximality of U." But why is this a contradiction ? If W_1, W_2 are the two *-comaximal elements, then U is contained in W = {V_1, ... , V_{n-1}, W_1, W_2} ? (I think you thought that U is contained in W, which contradicts the maximality of U. But U is not contained in W as a set.)

78.

81.  QUESTION (HD 1902): Given that * is a star operation of finite type. You call a *-finite *-ideal I homogeneous if I is contained in a unique maximal *-ideal in your paper with Dumitrescu in [JPAA, 214 (2010) 2087-2091] and you call I *-rigid if I is a finitely generated ideal that is contained in a unique maximal *-ideal in your Arxiv paper (I): https://arxiv.org/pdf/1712.06725.pdf. Are these the same concepts? Also in your Arxiv paper you call a maximal *-ideal M, potent if M contains a *-rigid ideal and in another Arxiv paper (II): https://arxiv.org/pdf/1802.08353.pdf you call M *-potent if M contains a *-homog ideal. Are they the same?

82.  QUESTION (HD 2001) "Fuchs in [On primal ideals, Proc. Amer. Math. Soc. 1 (1950), 1-6.] showed that every irreducible ideal is primal. I need an example to show the converse is not true." (This question was asked by Farimah Farokhpay of Shahid Chamran University, Ahvaz, Iran.)

83.  -homogeneous ideals" https://arxiv.org/pdf/1907.04384.pdf  Boy! What a mistake to make! I read your "explanation" after Theorem 2.3. Though I could not pinpoint the mistake , but isn't your admission-like explanation proof that you made a huge mistake?

84.  QUESTION (HD 2003) What is a "t-class group" of an integral domain D and how do you compute it?

88.  QUESTION (HD 2007) You write in your paper, [DZ, Comm. Algebra 39 (2011) 808--818] the following about Proposition 12: "The following result extends [28, Proposition 2.1]". Now, the above mentioned proposition is: "Let D be a t-Schreier domain and x₁,...,x_{n}D\{0} such that (x₁,...,x_{n})_{v}≠D. Then there exists a t-invertible t-ideal H such that (x₁,...,x_{n})H≠D." On the other hand the result [28, Proposition 2.1] is about sums of mutually disjoint homogeneous elements. Could you explain the connection? Similarly, I do not see any connection of [28, Proposition 2.1] with Proposition 13 of [DZ]. Is there an explanation?

89.  QUESTION (HD 2008) Someone has sent me a copy of a page from a Korean website                                     . This page has so graciously awarded a zero rating to my paper "On finite conductor domains" [Z, Manuscripta Math. 24(1978) 191-203]. How should I respond?

91.  QUESTION (HD 2102) I was reading the paper "On a general theory of factorization in integral domains", by Anderson and Frazier, Rocky Mountain J. Math., 41 (3) (2011), when I came across "In , the first author following suggestions of Zafrullah extended these notions to star operations." Now, what more does one want in terms of appreciation? On your part what I have seen lately, is complaints against Professor Anderson and attacks. Isn't it akin to biting the hand that feeds you?

93.                   QUESTION (HD 2104) The Wikipedia article     https://en.wikipedia.org/wiki/Schreier_domain  describes Schreier domains as integrally closed integral domains in which every nonzero element is primal, i.e., whenever x divides yz, x can be written as x = x_1 x_2 so that x_1 divides y and x_2 divides z. An integral domain is said to be pre-Schreier if every nonzero element is primal. The article also says that the term "pre-Schreier" was introduced by Muhammad Zafrullah. On the other hand the article https://planetmath.org/schreierdomain on planet Math calls pre-Schreier as a synonym of Schreier domains. Can you provide a reason why you introduced this new term?

94.               QUESTION (HD 2105) While reading your paper [J. Pure Appl. Algebra 212 (2008), 376--393] I got stuck at Lemma 3.7. Really, how do you propose to show that if an ideal A of R=D+XDS[X] is such that A/\S is non empty then A=(A/\D)R? Also, was it necessary to use A/\S is non empty in the proof of (JR)_{v}=(J_{v}R)_{v}?