This is my area of research, and I want to know everything that is new about it. You can send me your questions at email@example.com (or firstname.lastname@example.org ) 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.
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 email@example.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.
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 recently).
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 rejected from JPAA, without a proper referee report.
63. Semirigid GCD domains II (An expanded version of 62) (submitted circa 9-10-2020 (to appear in JAA) and later pre-published at Arxiv: https://arxiv.org/pdf/2012.10339.pdf
64. *-Almost super homogeneous ideals in *-h-local domains, (to appear in JAA) (With Shiqi Xing and Dan Anderson)
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, 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:
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 Paul’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.
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)
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)
and PVMD’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.
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 Nagata’s
theorem, J. Pure Appl. Algebra 86(1993), 109-124 (with D. Anderson
On Riesz Groups , Manuscripta Math. 80(1993), 225-238.
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).
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 Fontana’s homepage (General): http://mfontana-homepage.blogspot.com/
2. Marco Fontana’s homepage (Research): http://mfontana-homepage.blogspot.com/p/ricerca-research.html
3. Ayman Badawi’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/
Last Updated: June 13, 2005