Injective Modules and Injective Quotient Rings  book cover
1st Edition

Injective Modules and Injective Quotient Rings

ISBN 9781138401877
Published December 18, 2020 by CRC Press
120 Pages

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Book Description

Injective Modules and Injective Quotient Rings, in two parts, is the only book of its kind to combine commutative and noncommutative ring theory. This unique and outstanding contribution to the mathematical literature will immediately advance the studies of mathematicians and graduate students in the field.

Written by a leading expert in the field, Injective Modules and Injective Quotient Rings offers readers the key concepts and methods used in both noncommutative and commutative ring theory. Part I provides the first non-torsion-theory proof of the Teply-Miller theorem and the first statement and proof of the converse of the Teply-Miller-Hansen theorem. Many applications of these theorems to the structure of rings and modules are given, including generalizations of theorems of Cailleau-Beck and Matlis on the structure of ∑-injectives and commutative rings. Part II provides an alternative approach to the solution of Kaplansky's problem on the classification of FGC rings. Of particular importance is the consistent use of noncommutative ring theoretical techniques throughout Part II to obtain theorems lying purely in the domain of commutative ring theory.

Graduate students and mathematicians in both commutative and noncommutative ring theory will learn from the unique approach and new general methods in ring theory contained in Injective Modules and Injective Quotient Rings.

Table of Contents

PREFACE -- PART I INJECTIVE MODULES OVER LEVITZKI RINGS -- Abstract -- 1. Introduction -- 2. Annihilators and the Galois Connection -- 3. Levitzki Modules -- 4. Finite Annihilators -- 5. Sigma Quasi-injective Modules -- Appendix -- 6. Lemmas from Fitting-Krull-Schmidt -- 7. The Teply-Miller Theorem -- 8. A-Injective Modules -- 9. Kasch Rings -- 10. A-Rings -- Appendix -- 11. Injective Modules Over Nonnoetherian Commutative Rings -- 11.1 Introduction -- 11.2 Preliminaries -- 11.3 Proof of Beck’s Theorem -- 11.4 Commutative Sigma and Delta Rings -- 11.5 Artinian (Noetherian) Injectives Are Sigma (Delta) Injective -- 11.6 A-Polynomial Rings Are Polynomials Over A-Rings -- Problems -- Notes -- Acknowledgments -- References -- PART II INJECTIVE QUOTIENT RINGS OF COMMUTATIVE RINGS -- Abstract -- 1. Introduction -- 2. Survey of Relevant Background -- 3. Lemmas -- 4. Proof of Theorem B -- 5. Quotient-injective Pre-FPF Rings Are FPF -- 6. CFPF = FSI -- 7. FPF Rings with Semilocal Quotient Rings -- 8. FPF Rings with PF Quotient Rings -- 9. Note on the Genus of a Module and Generic Families of Rings -- 10. FP2F and CFP2F Rings and the "Big" Genus -- Problems -- References -- Abbreviations – INDEX.

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Carl Faith is Professor of Mathematics at Rutgers University, New Brunswick, New Jersey. He graduated from the University of Kentucky with honors in mathematics in 1951, and received his Ph.D. from Purdue University in 1955. In 1959-1960 he was NATO post-doctoral fellow at Heidelberg University. Professor Faith taught at several major academic institutions, including Purdue University, Michigan State University, and Pennsylvania State University, and he was also an NSF postdoctoral fellow and a member of the Institute for Advanced Study before he joined Rutgers in 1962. In 1970, he attended a portion of Tulane University's Algebra Year, and in 1965-1966 he was a visiting scholar at the University of California at Berkeley. He has lectured extensively in Europe and India. An author of numerous publications, including 5 books, Professor Faith's re­search interests are in ring theory, module theory, and Galois theory. He is a member of the American Mathematical Society and the Association of Members of the Institute for Advanced Study.