40th MEMORIAL EULOGY, MAY 16, 2001
by CARL FAITH
Appropriately my first contact with Professor Ahmad Shamsuddin was mathematical. In an 1982 paper by Ahmad (Ref [S1]) in the Canadian Math. Bulletin, I found a theorem that I needed to prove the main result in my 1987 paper (Ref. F) in the Canadian Journal of Math. I doubt that I would ever have found it on my own. Ahmad followed up on his results of [S1] in a 1982 paper [S2] in the Bulletin of the London Math. Soc. showing that a theorem, which Ahmad attributes to D.R. Lane, when k is algebraically closed, to the effect that every k-automorphism g of the polynomial ring k[x,y] in two variables leaves a nonzero proper ideal invariant, where k is an arbitrary field. Ahmad proved that this does not extend to the polynomial ring R = k[x,y,z] in three variables. In fact he takes g to be the automorphism sending x to x+1, y to y + xz + 1 and z to y + (x+1)z, where k can be any field of characteristic zero. This is a beautiful result and the proof requires a mastery of both technique and theory of commutative ring theory.
AHMAD AT RUTGERS (1992 – 1993)
So naturally we were delighted when Ahmad applied to take his 1992-1993 sabbatical from American University with us at Rutgers University in New Brunswick, New Jersey. Although Ahmad began his sabbatical at Rutgers on October 15th, 1992, we did not meet him until after Dolors Herbera arrived in late January 1993 for her Postdoctoral Fellowship from the Spanish Government. Early in February, I sent Ahmad a polite note asking him to join Dolors and me in a ring theory seminar, and gave him Dolors’ office number. I wanted my two visitors to meet and help prove the following conjecture that I had dreamed up:
(C) If M is a linearly compact module, then its endomorphism ring is semilocal.
The conjecture was based on the Camp’s – Dicks theorem [C – D] that states that C was true if M is Artinian. (This was proved in their 1993 paper before the conjecture (C) was dreamed up.)
AHMAD SHAMSUDDIN AND DOLORS HERBERA
When Ahmad contacted Dolors, they immediately began working on the conjecture, and, moreover, proving it without me in a matter of weeks using Camps-Dicks characterizations of semilocal rings, and the results of Hanna and Ahmad [Ha – S2] on dual Goldie dimension. (Also see [Ha – S1].) Eventually Ahmad and Dolors generalized conjecture (C) to prove that all modules having both finite Goldie and finite dual Goldie dimension have semilocal endomorphism rings. Their theorem, which appeared in [H – S 1] in the Proceedings of the American Math. Soc. in 1995, is what I called in [F3] a grand generalization of Schur’s lemma that dates back to his famous paper of 1904, almost a century ago! I personally think that these theorems will survive in mathematical thought as long as Schur’s will, i.e., forever.
This indicates the way mathematics develops — first Schur, then Hanna and Shamsuddin, then Camps and Dicks, and bingo, Herbera and Shamsuddin, each forging a strong chain of links.
HERBERA AND SHAMSUDDIN:
WHEN SELFINJECTIVE PERFECT RINGS ARE QF
A problem in Ring Theory is the conjecture of the author’s dating back to 1966:
(FC) If R is right and left perfect and right self-injective, then R is Quasi-Frobenius(=QF).
This was verified by B.L. Osofsky, a student and later a colleague of mine, assuming that in addition R is left self-injective. Another case determined by her is assuming that the Jacobson radical J of R is finitely generated modulo its square. See [F 3], [F-H], and [H – S 2] for background references.
Ahmad and Dolors proved the following partial verification of (FC):
HERBERA – SHAMSUDDIN THEOREM [H – S 2]
Conjecture (FC) holds under each of the following conditions:
(1) J modulo its square is countably generated as a left R-module
(2) The right annihilator of the intersection of two left ideals is the sum of their rightannihilators.
See [F – H] for an update on the (FC) conjecture.
VERIFICATION OF A CONJECTURE OF SHAMSUDDIN
Von Neumann Regular ( = VNR) rings are an important class of rings first introduced by Johnny Von Neumann in a paper in the Proceedings of the National Academy of Science back in 1931. During his visit to Rutgers University in September 1994, Ahmad made the following conjecture concerning VNR rings:
(SC) SHAMSUDDIN’S CONJECTURE. If R is a ring such that R modulo its Jacobson radical J is a finite product of simple VNR rings, then R is VNR iff R is a semiprime ring and J is an annihilator ideal.
In [F2] I verified (SC) by using a result of my paper [F1] which Ahmad was not aware of, and furthermore proved two additional characterizations of VNR rings which were inspired by Ahmad’s Conjecture. (See F).
I was quite happy to be able to solve Ahmad’s conjecture, but when I sent him a copy of my paper, he initially refused to allow me to refer to him in the title! However, I persuaded Dr. Carles Perello’, the editor of Publ. Math., that Ahmad was being unduly modest. The proof required some deep mathematics that went back to two papers by Utumi on VNR rings and other ideas involved in Shamsuddin’s conjecture (SC). Perello ruled that the title of a paper was in the domain of the author, and therefore let the title stand as it appeared. I was very happy to pay homage to Ahmad’s genius in this way, and in retrospect so was Ahmad.
AHMAD’S MATHEMATICAL GENEALOGY
Ahmad received his Ph.D. at Leeds in 1977 under the direction of Professor Robert Hart, and a part of his thesis appeared in the Journal of the London Math. Society in the same year. (See [SO].) Hart told us in an e-mail of May 11, 2001, that A.W. Goldie came to Leeds in 1963, and that he “learned a lot from him.” The same was true of Ahmad, which explains Ahmad’s deep knowledge of Goldie and dual Goldie dimension that was crucial in the proof of the now famous Herbera-Shamsuddin theorem.
So you see, mathematicians owe as much to their mathematical parents and grandparents, as everybody does to their natural parents and grandparents. My own great great mathematical grandfather, the immortal E.H. Moore, had 21 students and 4239 descendants, while my great mathematical grandfather, L.E. Dickson, had 53 students and 473 descendants. You will note how much more prolific mathematicians are in mathematical than in human progeny!
These things are very important in the development of mathematics. I first met Dolors Herbera at a mathematical seminar at Autonoma University of Barcelona back in spring 1986. The seminar was run by Dolors’ Ph.D. adviser, the late Pere Menal, who founded an outstanding school of Ring Theory at Autonoma. In the seminar, various problems arose, some raised by me, some by Pere, and some by others, many of which Dolors solved in her Ph.D. thesis. One significant problem that came up was the one that Camp and Dicks solved by their theorem mentioned above, and this led to the conjecture (C) above which Dolors and Ahmad solved.
SOME LEBANESE FRIENDS
I had known a number of Lebanese people before I met Ahmad in spring 1993. One was Sir Michael Atiyah, a professor at the Institute for Advanced Study (IAS) in 1969 – 1972, and other years when I visited IAS. IAS is democratic and everybody sits down together at lunch and tea. I have written about this at length in Part II of [F3] entitled “Snapshots of Mathematical People and Places.”
I met another noted Lebanese mathematician, Dr. Soumaya Makdisi-Khuri, for the first time in 1978, when I lectured at Yale. After my lecture, she and her late husband, Raja, invited me afterwards to stay in their cheerful house where I met their children, Fadlou, Rameses, and Janine. Raja, an MD, had been a Dean at American University before he accepted a position at Yale. Soumaya received her Ph.D. at Yale under Professor Charles Rickert and taught math at AUB for many years.
According to Souomaya, Fadlou Khuri was named after a Lebanese colleague of mine, Fadlou Shehadi, a philosophy professor at Rutgers University, and a marvelous basso opera singer in Lieder and Princeton Opera. I have known and esteemed Fadlou for almost forty years. (I found out about Fadlou’s namesake when I innocently asked Soumaya at Yale: “Do you know Fadlou Shedadi?”)
Another longtime Lebanese acquaintance, Lou Asack, is the owner of a very fine camera store in Princeton curiously called New York Camera and lives in my neighborhood on Riverside Drive.
Ahmad visited my wife Molly and me at our house in Princeton several times in Spring ’93 during his sabbatical, and also visited a favorite teacher of his from American University, Professor Kennedy of Princeton University. On one of these visits, Ahmad presented us with a beautifully illustrated “History of Islam.” Once Ahmad, Molly, Dolors and I went to the Metropolitan Museum of Art to see an exhibition of Islamic art. Ahmad praised everything as being “of the best.”
In spring 2000, when Ahmad underwent further treatment at Sloan-Kettering Hospital in New York, he brought Nisrine for two visits while he stayed at his sister’s Lubna’s house in Avenel. We met at “The Cafe” in Princeton where we talked for hours, once with our son Malachi who had known Ahmad in spring 1993, and Malachi’s recent bride, Jhilam, originally from Bangladesh. Inasmuch as Malachi had converted to Islam, there were four Moslems, a Unitarian and a Baptist (me) at our luncheon table! We had great fun. Despite his post-operative discomforts, Ahmad maintained his legendary sense of humor throughout. We were dazzled by Nisrine’s notable beauty, intelligence, and her linguistic ability. Her love for Ahmad, and his for her, were in evidence in every gesture and glance that they made. They absolutely adored each other, and in his last letter to me, Ahmad called her his “angel.”
Ahmad loved to travel, and after I knew him, he sent me postcards, from an ancient Crusader fortress on the Mediterranean that he visited with Nisrine, and from Paris and London, all of which I kept, as well as his many letters describing his research and personal matters. Besides Rutgers Ahmad took a sabbatical, or semi- sabbatical, at the University of California, Berkeley in 1988 – 9. (He wrote me about this when I sent him postcards from Berkeley which I visit twice a year for two or three days.) Another time he returned to Leeds for a semester, stopping off in London. Once, he visited Vic Camillo at the University of Iowa, Iowa City, where his sister, Hala, a nurse, resides. (Hala told me her name means “thanks,” so named because she was the first daughter after four sons!) Another time he visited S.K.Jain at Ohio University in Athens. While undergoing treatment at Sloan-Kettering Hospital in New York in spring 2000, Ahmad was a visitor at the Courant Institute.
Whenever Ahmad came to the USA, which was frequently, he would call me up and we would have long conversations, sometimes lasting an hour or more. I used to joke, “Ahmad, you never have to stay in a hotel!” because he had relatives all over, and he would laugh helplessly and agree. Besides his sisters in Iowa and New Jersey, he would stay with the youngest of his four brothers in Chicago and an uncle, who worked for the World Bank, in Washington, D.C. I still half expect to hear from him. As Marcel Proust once wrote about death: “People do not die immediately for us, but remain in an aura of life…as if they are traveling abroad.” I like to think that of Ahmad.
We were honored by the invitation of Nisrine Shamsuddin and Professor Wadi Jereidini of the Department of Mathematics to be asked to speak at Ahmad’s 40th Memorial. We regret that for reasons of health, especially a surgical procedure thatwe underwent on April 26th we were unable to come to Beirut to pay our respects at this time. I wish to thank my own angel, my wife, Molly Sullivan, for putting this eulogy into type and on the net. I regret that I have been unable to obtain a copy of Ahmad’s CV from American University of Beirut.
In his last letter to me (e-mail of March 25), Ahmad said that he had asked the American Mathematical Society to help him establish a fund to encourage mathematics in Lebanon. Pursuant to his wishes we contacted Dr. John Ewing, Executive Vice-President of the AMS, who upon reflection has made a number of suggestions which have been forwarded to Professor Jureidini. The specifics will be decided and announced at a later date. In addition Professor Mark Teply, Executive Editor of Communications in Algebra has kindly agreed to collect and expedite papers dedicated to Ahmad into a single volume dedicated to him.
In memory of a loving and courageous friend, Dr. Ahmad Shamsuddin, and to his courageous and loving widow, Nisrine Shamsuddin. have also dedicated my paper [F4] to Ahmad and Nisrine.
[C – D] R. Camps and W. Dicks, On semilocal rings, Israel J. Math. 8l1 (1993), 203 – 211.
[F 0] C. Faith, On the Galois theory of commutative rings, II: automorphisms induced in residue rings, Canad. J. Math. 39 (1987) 1025 – 1037.
[F1] C. Faith, The Maximal regular ideal of self-injective and continuous rings splits-off, Arch.Math.4 (1985), 511-521.
[F2] C. Faith, New characterizations of von Neumann regular rings and a conjecture of Shamsuddin, Publ. Mat. 40 (1996), 383-385.
[F3] C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Surveys of the Amer. Math. Soc. vol. 65, Providence, 1999.
[F4] C. Faith, Coherent rings and annihilator conditions in matrix and polynomial rings, in Handbook of Algebra, vol. 3, M. Hazewinkel (ed.), Elsevier B. V., Amsterdam 2002.
[F – H] C. Faith and D.V. Huynh, When selfinjective rings are QF: A report on a problem, J. Algebra and Applications, 1(1), 2002.
[H – S 1] D. Herbera and A. Shamsuddin, Modules with semilocal endomorphism ring, Proc. Amer. Math. Soc. 128 (1995) 3593 – 3600.
[H – S 2] D. Herbera and A. Shamsuddin, On self-injective perfect rings, Canad. Math. Bull. 39 (1996) 55 – 58
[Ha – S1] A. Hanna and A. Shamsuddin, Duality in the Category of Modules. Applications, Algebra Berichte 49, Verlag Reinhart – Fisher, 1984.
[Ha – S2] A. Hanna and A. Shamsuddin, Dual Goldie dimension, Rend. di Mat. U. Trieste, 24 (1992) 25 – 38.
[S 0] Ahmad Shamsuddin, A note on a class of simple Noetherian rings, J. London Math. Soc., 15 (1977) 213-16.
[S1] Ahmad Shamsuddin, Rings with automorphisms leaving no nontrivial proper ideals invariant, Canad. Math. Bull., 25 (1982) 478 – 486.
[S2] Ahmad Shamsuddin, On automorphisms of polynomial rings, Bull. London Math. Soc. 14 (1982), 407 – 409.
[Sc] I. Schur, Ueber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionem, J. reine u.angew.Math.127 (1904) 20 – 50