Background
Faith, Carl Clifton was born on April 28, 1927 in Covington, Kentucky, United States. Son of Herbert Spencer and Vila Belle (Foster) Faith.
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book
FPF Ring Theory: Faithful Modules and Generators of Mod-R (London Mathematical Society Lecture Note Series)
(This is the first book on the subject of FPF rings and th...)
This is the first book on the subject of FPF rings and the systematic use of the notion of the generator of the category mod-R of all right R-modules and its relationship to faithful modules. This carries out the program, explicit of inherent, in the work of G Azumaya, H. Bass, R. Dedekind, S. Endo, I. Kaplansky, K. Morita, T. Nakayama, R. Thrall, and more recently, W. Brandal, R. Pierce, T. Shores, R. and S. Wiegand and P. Vamos, among others. FPF rings include quasi-Frobenius rings (and thus finite rings over fields), pseudo-Frobenius (PF) rings (and thus injective cogenerator rings), bounded Dedekind prime rings and the following commutative rings; self-injective rings, Prufer rings, all rings over which every finitely generated module decomposes into a direct sum of cyclic modules (=FGC rings), and hence almost maximal valuation rings. Any product (finite or infinite) of commutative or self-basic PFP rings is FPF. A number of important classes of FPF rings are completely characterised including semiprime Neotherian, semiperfect Neotherian, perfect nonsingular prime, regular and self-injective rings. Finite group rings over PF or commutative injective rings are FPF. This work is the culmination of a decade of research and writing by the authors and includes all known theorems on the subject of noncommutative FPF rings. This book will be of interest to professional mathematicians, especially those with an interest in noncommutative ring theory and module theory.
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Algebra II Ring Theory: Vol. 2: Ring Theory (Grundlehren der mathematischen Wissenschaften)
(Book by Faith, Carl)
Book by Faith, Carl
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Algebra: Rings, Modules and Categories I (Grundlehren der mathematischen Wissenschaften)
(VI of Oregon lectures in 1962, Bass gave simplified proof...)
VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya 511 and Auslander-Goldman 60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction.
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Algebra: Rings, modules and categories (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, No. 190)
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Algebra II: Ring Theory (Grundlehren der mathematischen Wissenschaften 191) (Vol 2)
(From Math Reviews: This is the second in a two volume set...)
From Math Reviews: This is the second in a two volume set. It is composed of ten chapters labeled 17 through 26 (the first 16 chapters are in Volume 1). Chapter 17 studies the lattice of submodules of a module over an associative ring, with such theorems as the Jordan-Hölder, Fittings, and Levitzkis'. Chapter 18 begins by introducing the Jacobson radical of a module, small submodules, decomposition into direct sums à la Remak, Krull, Schmidt and finishes with semi-perfect rings. The material in Chapter 19 concerns quasi-injective modules and their endomorphism rings. It is in this setting that the Jacobson density theorem is given, von Neuman regular rings, rational extensions and maximal quotient rings are introduced, and the Goldie theorems are proved. QI rings are also studied in this chapter. Chapter 20 begins with the theorems on how the chain conditions relate to the cardinality of the summands of direct sum decompositions of injectives. Next Chatter's theorem on the decomposition of Noetherian rings into semiprime and Artinian rings is given. The chapter concludes with a section on valuation and almost maximal valuation rings. Chapter 21 deals with direct sums of modules, each of which has a local endomorphism ring, hence Azumaya type uniqueness theorems on summands of infinite direct sums of indecomposable modules. Chapter 22 is devoted to perfect rings. The subject of dualities between module categories is extensively studied in Chapter 23. The basic results on quasi-Frobenius rings are given in Chapter 24. Chapter 25 begins with Warfield's structure theorem of serial rings, then gives Nakayama's structure theorem for Artinian serial rings (generalized uniserial). Next is a look at rings for which every finitely generated module is the direct sum of cyclic modules and finally the Eisenbud-Griffith-Robson theorem. Chapter 26 contains the classical descriptions of the prime and Jacobson radicals. It also contains Amitsur's theorem.
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Lectures on Injective Modules and Quotient Rings (Lecture Notes in Mathematics)
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Lectures on Injective Modules and Quotient Rings (Lecture Notes in Mathematics) by Carl Faith (2009-02-22)
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Algebra: Rings, modules and categories I. (Grundlehren der mathematischen Wissenschaften, Bd.190) (Grundlehren Der Mathematischen Wissenschaften) by Carl Clifton Faith (1981-07-01)
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Carl Clifton Faith
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Carl Clifton Faith, American Mathematics educator. National Science Foundation grantee; North Atlantic Treaty Organization postdoctoral fellow Heidelberg, 1959-1960;l National Science Foundation postdoctoral fellow, 1960-1961; Rutgers faculty fellow, 1965-1966, 69-70; academic sabbatical, 1973-1974, fall 1977, 89, spring 1982, 86; research grantee Ministry of Education, Spain, 1989-1990, Italy, 1991.
Education
Student, University Cincinnati, 1947; Algernon Sidney Sullivan fellow, U. Kentucky, 1949-1951; Bachelor of Science cum laude with honors in Mathematics, U. Kentucky, 1951; Master of Arts in Mathematics, Purdue University, 1953; Doctor of Philosophy in Mathematics, Purdue University, 1955.
Career
Assistant professor mathematics, Michigan State University, 1955-1957;
assistant, associate professor, Pennsylvania State University, 1957-1962;
professor mathematics, Rutgers University, since 1962. Visiting member Institute Advisory Study, Princeton, New Jersey, 1960-1962, 73-74, 77-78, associate member, since 1983. Visiting professor U. Heidelberg (Germany), 1959-1960, U. New Mexico, Las Cruces, 1982, Centre Recerca Matematica Universitat Autonoma, Barcelona, Spain, Institut d'Estudis Catalans, Barcelona, spring 1986, summer 1987, fall 1989.
Visiting scholar University of California, Berkeley, 1965-1966. Consultant Agency for International Development, National Science Foundation, India, 1968. Screening committee Senior Fulbright awards, 1970-1973.
Achievements
Works
book
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FPF Ring Theory: Faithful Modules and Generators of Mod-R (London Mathematical Society Lecture Note Series)
(This is the first book on the subject of FPF rings and th...)
-
Algebra: Rings, Modules and Categories I (Grundlehren der mathematischen Wissenschaften)
(VI of Oregon lectures in 1962, Bass gave simplified proof...)
-
Algebra II: Ring Theory (Grundlehren der mathematischen Wissenschaften 191) (Vol 2)
(From Math Reviews: This is the second in a two volume set...)
-
Algebra II Ring Theory: Vol. 2: Ring Theory (Grundlehren der mathematischen Wissenschaften)
(Book by Faith, Carl)
- Algebra: rings, modules and categories (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen)
- Algebra: Rings, modules and categories (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, No. 190)
- Lectures on Injective Modules and Quotient Rings (Lecture Notes in Mathematics)
- Lectures on Injective Modules and Quotient Rings (Lecture Notes in Mathematics) by Carl Faith (2009-02-22)
- Algebra: Rings, modules and categories I. (Grundlehren der mathematischen Wissenschaften, Bd.190) (Grundlehren Der Mathematischen Wissenschaften) by Carl Clifton Faith (1981-07-01)
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FPF Ring Theory: Faithful Modules and Generators of Mod-R (London Mathematical Society Lecture Note Series)
Membership
Block captain Princeton Association Human Rights, since 1963. Leadership group Aminesty International, since 1991. Served with United States Naval Reserve, 1945-1946.
Member American Mathematics Society, Mathematics Association American, Canadian Mathematics Society, London Math Society, Japanese Mathematics Society.
Connections
Married Betty Frances Compton, August 11, 1951 (divorced April 28, 1981). Children: Heidi, Cindy. Married Molly Kathleen Sullivan, September 10, 1987.
Stepchildren: Zeno Wood, Japheth Wood, Malachi Wood, Ezra Wood.
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Born
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Nationality
American
Awards
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National Science Foundation grantee. North Atlantic Treaty Organization postdoctoral fellow Heidelberg, 1959-1960;l National Science Foundation postdoctoral fellow, 1960-1961. Rutgers faculty fellow, 1965-1966, 69-70. Academic sabbatical, 1973-1974, fall 1977, 89, spring 1982, 86. Research grantee Ministry of Education, Spain, 1989-1990, Italy, 1991.National Science Foundation grantee. North Atlantic Treaty Organization postdoctoral fellow Heidelberg, 1959-1960;l National Science Foundation postdoctoral fellow, 1960-1961. Rutgers faculty fellow, 1965-1966, 69-70.
Academic sabbatical, 1973-1974, fall 1977, 89, spring 1982, 86. Research grantee Ministry of Education, Spain, 1989-1990, Italy, 1991.