Questions Posted

  1. QUESTION (HD0301): If R=F[X³,X^{4}],I=(X⁶,X⁷,X⁸) then how is I⁻¹=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 P⁻¹=R?
  9. QUESTION (HD0309): If D is an integral domain, and M a prime ideal of D[X] with M∩D=(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. (This question was asked by Jesse Elliott of CSU Channel Island.)