This is my area of research, and I want to know everything that is new about it. You can send me your questions at zafrullah@lohar.com (or mzafrullah@usa.net ) I will display them for everyone to see and comment. (Some questions are never asked because they occurred to a person who is shy, or is worried about the answer being common knowledge. Ask away the questions now. I will display your question, at your request, as anonymous. Once I know that the experts think your question is a valid question, and that answers to it will increase our knowledge content, I will include your name and address and invite you to make some comments on how to answer this question.) If your work is related to multiplicative ideal theory, and new, I want everyone to know about it. You can send me the title and abstract of your Ph D thesis (if you are a new Ph D), and you can send me the title, abstract and other relevant data of your upcoming or recently published paper. Whatever you send me will be displayed under a suitable heading for everyone to see. If you need to make an announcement about an upcoming conference or if you want to make a call for papers on multiplicative ideal theory, just send me the relevant material in a presentable form. I will display it.
Some Papers (Recent
and Old)
Recent
Listed below are some of my recent paper and talks. Click on the highlighted part to get a pdf version of the preprint or talk. Please remember, I am just a human. So, if you find some error /errors, in what I have put here, do please let me know at zafrullah@lohar.com. I would not hide my appreciation of this kindness, as some of you know. Here then are some articles, click away at what intrigues you or what you have not yet seen.
3. Distinguished domains, Internat. J. Comm. Ring Theory (to appear) (with D.D. Anderson and J. Kwak).
4. LCM-splitting sets in some ring extensions Proc. Amer. Math. Soc.(to appear),(with T. Dumitrescu).
6. Various Facets of Rings between D[X] and K[X]. (Survey Article ) Comm. Algebra (to appear).
7. A Property of weakly Krull domains, Proc. Amer. Math. Soc.(to appear) (with D.D. Anderson).
10. A note on almost GCD monoids, Semigroup Forum( to appear) (with D.D. Anderson)
11. Weakly Krull inside factorial domains, Proceedings of the Chapel Hill Conference (to appear) (with D. D. Anderson and G. W. Chang)
12. UMV-domains,
Proceedings of the Chapel Hill Conference (to appear) (with E.G.
13. Factorization of certain sets of polynomials in an
integral domain, Internat. J. Comm. Ring
Theory (with D. D. Anderson and Pramod K. Sharma)
14. t-Splitting multiplicative sets of ideals in integral domains, J. Pure Appl. Algebra (to appear) (with G. W. Chang and T. Dumitrescu)
15. Domains over which polynomials split (preprint)
16. The w-integral closure of integral domains, with G. W. Chang (to appear in J. Algebra)
17. The Schreier property and Gauss' Lemma, with D. D. Anderson (to appear)
18. The half-factorial property and the domains of the form A+XB[X], with J. Coykendall and T. Dumitrescu (to appear in Houston J. Math.)
19. What v-coprimality can do for
you. A talk given at a workshop, on
Commutative Rings and their Modules, held at
20. Quasi-Schreier domains II, Comm. Algebra (to appear) with D. D. Anderson and Tiberiu Dumitrescu
21. What v-coprimality can do for you, a survey article, to appear in " Multiplicative Ideal Theory in Commutative Algebra: A tribute to the work of Robert Gilmer", Jim Brewer, Sarah Glaz, William Heinzer, and Bruce Olberding Editors, Springer.
22. Pseudo Almost Integral elements, Comm. Algebra (to appear) with D.D. Anderson
23. Monoid Domain Constructions of antimatter domains, Comm. Algebra (to appear) with D.D. Anderson, J. Coykendall, and Linda Hill
24. Factoriality in Riesz groups, with Joe L. Mott and Muneer A. Rashid (to appear in J. Group Theory)
25. Unique Representation domains, II, with Said El Baghdadi and Stefania Gabelli (to appear in J. Pure Appl. Algebra)
26. Some remarks on Prufer *-multiplication domains and class groups, with David Anderson and Marco Fontana, (to appear in J. Algebra)
27. A Question/Answer session on v-domain. A kind of survey on v-domains, that includes a bit of history (recent and old) of v-domains, some description of star operations and ideal systems and most of known results on v-domains. The mode is Question/Answer and relaxed. You can write to the author for more or to correct him or to suggest improvements.
28. Almost Bezout domains, III44444 (with Dan Anderson) to appear in Bull Math Soc Sci Math Roumanie.
29. Some Applications of ZornÂ’s Lemma in Algebra, (with Dan Anderson and David Dobbs) to appear in Tamkang J. Math.
30. On v-domains and star operations, (with D.D. Anderson, D.F. Anderson and M. Fontana) to appear in Comm. Algebra.
31. On v-domains: a survey, (with Marco Fontana), accepted for “Recent Developments in Commutative Algebra”
32. Splitting sets and weakly Matlis domains, (with Dan Anderson) accepted for “Fez Proceedings”
33. -t-invertibility and Bazzoni-like Statements, submitted to Journal of Pure and Applied Algebra, on April 6, 2009, accepted for publication on July 2, 2009.
34. Integral domains in which nonzero locally principal ideals are invertible, Comm. Algebra, (to appear) (with Dan Anderson).
35. A “v-operation free” approach to Prufer v-multiplication domains, IJMMS, (to appear) (with M. Fontana).
36. t-Schreier domains, Comm. Algebra, (to appear) (with T. Dumitrescu)
37. Characterizing domains of finite *character, J. Pure Appl. Algebra (to appear) (with T. Dumitrescu). This paper is a modified form of the original manuscript submitted to JPAA.
38. Characterizing domains of finite *-character (original submitted version)
39. *-Finite ideals contained in infinitely many *s-maximal ideals (to appear) (with Dan Anderson)
40. The v-operation in extensions of integral domains, J. Algebra Appl., (to appear) (with David Anderson and Said El Baghdadi)
41. Bases of pre-Riesz groups and ConradÂ’s F-condition, (to appear) (with Y.C. Yang)
42. Factorization and the A + XB[X] construction, (Copy of a talk that I gave at Iowa Commutative Algebra Conference held at the University of Iowa, Iowa City, IA in March 2011.)
43. Star operations of finite character induced by rings of fractions, (I was looking through some old mail when I came upon a script that I sent Marco Fontana in 2006. I sent it to him as a proposed project. In the hope that it may be of help to someone, I am posting it. It was a sort of hurried affair so it may contain some errors. It may contain some concepts that I proposed to study, then but never came around to doing that. One of them is the notion of almost PVMD a notion that was later introduced and studied independently by Li Qing, a student of Fanggui WangÂ’s.) Note added 9-25-2012. It so happens that Rebecca Lewin had introduced APVMDs in her doctoral dissertation which is available online at: http://ir.uiowa.edu/cgi/viewcontent.cgi?article=3252&context=etd I do stand corrected. However Li Qing and some subsequent contributors, such as G.W. Chang, H.K. Kim and J.W. Lim have gone way beyond introducing the concept thanks to the work done after RebeccaÂ’s graduation.
44. A+XB[X] construction and the pre-Schreier property
45. Integral domains of finite t-character , J. Algebra (to appear) (with D.D. Anderson and G.W. Chang)
46. Corrigendum to “integral domains of finite t-character” J. Algebra (to appear) (With D.D. Anderson and G.W. Chang)
47. Integral domains in which any two v-coprime elements are comaximal J. Algebra (to appear) (with E.G. Houston)
48. Locally GCD domains and the ring D+XD_S [X], (with Chang and Dumitrescu)
49. Nagata-like theorems for integral domains of finite character and finite t-character , J. Algebra and Applications (to appear) (with D.D. Anderson and G.W. Chang)
50. Cohen-type theorems for a commutative ring, Houston J. Math. (to appear) (With D.D. Anderson)
51. On locally AGCD domains, J. Algebra and Appl. (to appear) (With D.F. Anderson and G.W. Chang)
52. Completely integrally closed Prufer v-multiplication domains , Comm. Algebra (to appear) (With D.F. and D.D. Anderson)
53. Graded Prufer domains , Comm. Algebra, (to appear) (With D.F. Anderson and G.W. Chang)
54. On *-Power conductor domains, (Submitted) (With D.D. Anderson and Evan G. Houston)
55. *-Superpotent domains, J. Commut. Algebra (to appear) (With Evan Houston)
56. A unified theory of domains leading to UFDs, On star-SH domains,
57. A more recent version of 56, above, written as a joint paper with Dan Anderson, of UIOWA: On *-semi homogeneous domains (with Dan Anderson, who has produced a wonderful new version: On star semi homogeneous domains that has been accepted for inclusion in the upcoming Springer’s David Anderson volume. Writing this version has caused some corrections and tweaks in 57. I thank Dan for insisting on re-writing. Now while Dan’s version follows the basic theory of 57 there are differences from 57 and from the arxiv version of 57. Click here for a tweaked version of https://arxiv.org/abs/1802.08353 )
58. On S-GCD domains, J. Algebra and Applications (to appear) (with D.D.Anderson and A. Hamed)
59. $t$-local domains and valuation domains, “Advances in Commutative Algebra” Ayman Badawi and James Barker Coykendall, Editors, Springer (to appear) (with Marco Fontana)
60. Domains whose ideals meet a universal restriction, (submitted ). It was first submitted to comm. algebra. The referee appeared to like the contents, but called it a survey, perhaps because of the applications of the idea. The handling editor saw his first chance of rejecting my paper and grabbed it saying that they were not accepting surveys. Instead of fighting it out there, I added another section and submitted it to J. Algebra and Applications. Now that itÂ’s over six months without a comment I thought I should at least put the new version at my web page. So here it is, with the same title but a different link New version
61. Star potent domains and homogeneous ideals, submitted (on March 18, 2020) to be considered for inclusion in Springer’s Dan Anderson volume, it has been accepted for inclusion in the volume.
62. Comments on unique factorization in non-unique factorization domains, submitted and got rejected without a proper referee report from JPAA.
63. Semirigid GCD domains II (An expanded version of 62.)
65. Revisiting G-Dedekind domains
66. Riesz and Pre-Riesz monoids
67. Revisiting G-Dedekind domains (latest)
68. On star homogeneous ideals
69. My comments on D+M constructions with general overrings by Brewer and Rutter (Longer annotated comments may be hard to read because of navigation problems. So, I am including the longest comment here: It appears that most of the results from the Costa-Mott-Zafrullah paper, that is referenced as [4], stated for T=D+XK[X] have been expanded on without mentioning the source results, even as Corollaries. I do not deny the insight that went into the Brewer-Rutter results, but I object to the manner in which the results from [4] were adopted without due reference to the special cases. The result is that folks reference this paper, rather than [4]., even for D+XK[X] situations. That is why, I feel within my rights to suggest that reference [4] that appeared, much later, as [J. Algebra 53(1978) 423-439] should be mentioned when this paper is mentioned. I also suggest that J. Algebra records should disclose the name of the referee of reference [4], as it appears the referee "sat" on the paper ([4]) to make sure that reference [4] was sufficiently scuttled. Finally I thank God that Brewer and Rutter lacked the intellectual girth to see what the D+XD_S[X] construction could do.) Let me also note that in the references of the Brewer-Rutter paper [4] is indicated as “submitted”. That means they had the full content of the paper at their disposal.
70. Well-Behaved prime t-ideals and almost Krull domains (with Evan Houston).
71. Integral domains and the idf property (with Felix Gotti). (If my website’s link does not work, check this out: https://arxiv.org/pdf/2203.02640.pdf )
72. On Chevalley’s Extension Theorem
73. On Chevalley’s Extension Theorem (latest)
74. Zafrullah’s flat (overring) problem
75. Condensed domains and the D+XL[X] construction
Some important papers in
Multiplicative Ideal Theory
Here are some classic papers whose authors have given me permission to put them on display.
D. D. Anderson, Star-operations induced by overrings, Comm. Algebra 16(12)(1988) 2535-2553.
D.D.
Anderson, Non-atomic unique factorization
domains, Non-atomic unique factorization in integral
domains, in Properties of Commutative Rings and Modules (ed. S. Chapman) SRC
Press Boca Raton, (2005), 1-21. It is an interesting account of the history of
the study of unique factorization in “non-unique factorization domains”. Pretty accurate. However, in this paper, he (Dan)
mentions generalized UFDs (GUFDs) and mentions [Anderson, Anderson and Zafrullah, Bollettino U. M. I.
(7) 9-A (1995), 401-413.] as their source. The fact is that GUFDs were
studied in the first chapter of my doctoral dissertation that I submitted to
the University of London in 1974. The dissertation can be downloaded from here:
https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.704293
A
copy of the dissertation is available on this page too. The point is that if
the property of unique factorization in integral domains that are non-unique
factorization domains is interesting enough that papers on this topic appear in
high level journals, then the topic started long ago, in
1974. Only “Muhammad Zafrullah”
could not publish his work and had to seek joint authorship with the Anderson
Brothers, who obliged and Muhammad Zafrullah is
grateful for their kindness.
D.F. Anderson and D.E. Dobbs, Pairs of rings with the same prime ideals, Can. J. Math. 32(2)(1980) 362-384.
J. Brewer and W. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1974), 1-7. While the associated primes of principal ideals were defined and used prior to this paper the treatment of them by these authors has had me go to this paper, time and again. .
P.M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64(1968), 251-264. (This paper forged a new direction in the study of divisibility and factorization by introducing the notion of a Schreier domain as an integrally closed integral domain with group of divisibility a Riesz group.) Paul Cohn gave me a reprint and I am proud to have kept it since 1972 and read it. I must confess that I have just scratched the surface, there is much more to that paper than just Schreier domains. To see some of the effects of the Schreier domain part look up The Schreier property and Gauss' Lemma, (this paper incidentally has appeared: Bolletino U.M.I. 8 10-B (2007), 43-62.).) I could not get PaulÂ’s permission to put this paper on my webpage, as I always thought I would get a good copy and then ask for his permission, it never occurred to me that he could die. In any case I wrote to Cambridge University Press, did not hear from them, except for an acknowledgment. Now I am putting it up with the intention that if there is an objection I would take down the paper and keep up the reference. The world is too technical for my taste.
J. Hedstrom and
Isidore Fleischer, Abstract Prufer Ideal Theory, J. Algebra 115(2)(1988) 332-337 and Errata on two articles (J. Algebra 124(1989) 533.
Marco Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. 123(4) (1980), 331--355.
Marco Fontana and Salah Kabbaj,
On
the Krull and valuative
dimension of D+XD_S [X] domains, J. Pure and Applied Algebra
63(1990), 231-245.
Laszlo Fuchs, Riesz Groups, Ann. Scoula Norm. Sup.
Some Old Papers (of
which I have very few reprints, if any)
 A note on two generator finite groups with two relations
Unique factorization and related topics, doctoral thesis (University of London, 1974).
On a result of Gilmer, J. London Math. Soc. (2)(1977), 19-20.
On finite conductor domains, Manuscripta Math. 24(1978), 191-204.
The construction D + XD_S [X], J. Algebra 53(2)(1978), 423-439 (with Douglas Costa and Joe Mott)
On Prufer v-multiplication domains, Manuscripta Math., 35(1981), 1-26 (with Joe Mott)
A general theory of almost factoriality, Manuscripta Math., 51(1985), 29-62. (The unpublished paper of mine cited as “The *-property” later appeared as: On a property of pre-Schreier domains)
The GCD property and quadratic polynomials, J. Math. 9(1986), 749-752 (with S.B. Malik and J.L. Mott).
On generalized Dedekind domains, Mathematika 33(1986), 285-296.
On a property of pre-Schreier domains, Comm. Algebra 15(1987), 1895-1920.
On some class groups of an integral domain, Bull. Soc. Math. Grece (N.S.) 29(1988), 45-59 (with A. Bouvier).
The D+XD[1/S][X] construction from GCD domains, J. Pure Appl. Algebra 50(1988), 93-107.
Two characterizations of Mori domains, Math. Japonica 33(4)(1988), 645-652.
Ascending chain conditions and star operations, Comm. Algebra 17(6)(1989), 1523-1533.
Some characterizations of v-domains and related properties, Colloq. Math. Vol. LVIII (1989), 1-9 (with D.D. Anderson, D.F. Anderson, D. Costa, D. Dobbs and J.L. Mott).
Some quotient based statements in Multiplicative Ideal Theory, Bollettino U.M.I. (7) 3-B (1989), 455-476 (with Anderson and Mott).
On t-invertibility II, Comm. Algebra 17(8)(1989), 1955-1969 (with Evan Houston)
t-linked overrings
and PVMDÂ’s, Comm. Algebra 17(11)(1989), 2835-2852 ( with D. Dobbs,
Flatness and invertibility of an ideal, Comm. Algebra 18(7)(1990), 2151-2158
Contents of polynomials and invertibility, Comm. Algebra 18(1990), 1569-1583 (with J. Mott and B, Nashier)
Factorization in integral domains, J. Pure Appl. Algebra 69(1990), 1-19 (with Dan Anderson and David Anderson). (Fact2) *see note below.
t-linked overrings
as intersections of localizations, Proc. Amer. Math. Soc. (1990),
637-646 (with D. Dobbs,
Well behaved prime t-ideals, J. Pure Appl. Algebra (1990), 199-207
Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109(4)(1990), 907-913 (with Dan Anderson)
Rings between D[X] and K[X], Houston J. Math. 17(1)(1991), 109-129 (with D.D. Anderson and D.F. Anderson). (Fact1) *see note below.
Almost Bezout domains, J. Algebra 142(2)(1991), 285-309 (with Dan Anderson)
Pseudo-Integrality, Canad. Math. Bull. 34(1)(1991), 15-22
(with D.F. Anderson and E. Houston).
Splitting the t-class group, J. Pure Appl. Algebra 74(1991), 17-37 (with D.D. Anderson and D.F. Anderson)
Finite character representation for integral domains, Bollettino U.M.I. (7) 6-B (1992) 613-630. (with D.D. Anderson and J. Mott)
On t-linked overrings,
Comm. Algebra 20(5)(1992), 1463-1488 (with D. Dobbs,
Factorization in integral domains, II, J. Algebra 152(1)(1992), 78-93 (with Dan Anderson and David Anderson). (Fact3) *see note below.
t-linked extensions, the t-class group, and
NagataÂ’s theorem, J. Pure Appl. Algebra 86(1993), 109-124 (with D.
Anderson and
On Riesz Groups , Manuscripta Math. 80(1993), 225-238.
t-invertibility
and comparability, Commutative Ring Theory (eds. P.-J. Cahen, D. Costa,
M. Fontana and S.-E. Kabbaj), Marcel Dekker,
P.M. Cohn's completely primal elements Zero-Dimensional Commutative Rings (eds. D.F. Anderson and D. Dobbs) Marcel Dekker, New York, 1995, 115-123 (with D.D. Anderson).
Examples in Modern Algebra with
which students can play, Primus 6 No.
4, (1996), 351-354 (with T. Jackson).
Putting t-invertibility
to use, Chapter 20, in Non-Noetherian Commutative Ring Theory, 429--457, Math.
Appl., vol. 520, Kluwer Acad. Publ.,
The Basis of Modern day Factorization Theory: The papers with Dan and David Anderson marked with (Fact 1), (Fact 2), (Fact 3) were written and submitted in that order. The paper marked (Fact 2) appeared earlier but it refers to (Fact 1) indicating that (Fact 1) was to appear. Whatever the order, these papers formed the basis of what is called the Factorization Theory. Some authors prefer to use “non-Unique Factorization” for what I call Factorization, but I have serious misgivings about that. My point is “Factorization” includes Unique Factorization, where the ideas of the Factorization Theory evolved from, whereas the term “non-Unique Factorization” obviously shuns Unique Factorization. It is remarkable though that the non-Unique Factorization crowd often end up using Unique Factorization in one form or another. The notion of “factorization in integral domains” was translated into the monoid setup by Franz Halter-Koch and Alfred Geroldinger. For their early work see [Factorization in integral domains, edited by D.D. Anderson, Lecture Notes in Pure and Applied Mathematics, Volume 189, Dekker, 1997]. Also, see the book by Alfred Geroldinger and Franz Halter-Koch [non-Unique Factorizations: Algebraic Combinatorial and Analytic Theory, Chapman & Hall/CRC 2006]. These two gentlemen and their group at Karl Franzen University, Graz, Austria have greatly expanded the scope of Factorization.)
Some
Useful/Important links
1. Marco FontanaÂ’s homepage (General): http://mfontana-homepage.blogspot.com/
2. Marco FontanaÂ’s homepage (Research): http://mfontana-homepage.blogspot.com/p/ricerca-research.html
3. Ayman BadawiÂ’s
homepage: http://www.ayman-badawi.com/
4. Multiplicative Ideal Theory Help Desk: http://www.lohar.com/mithelpdesk/
(At this link you will find
answers to some recently asked questions about topics in Multiplicative ideal
Theory. Send in a question, if you have one.)
5. Commalg.org Surveys: https://www.commalg.org/surveys/
6. Commalg.org home: https://www.commalg.org/
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Last Updated: June 13, 2005