Injective Modules and Injective Quotient Rings

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.

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.