Power series provide a technique for constructing examples ofcommutative rings. In this book, the authors describe this techniqueand use it to analyse properties of commutative rings and theirspectra. This book presents results obtained using this approach. Theauthors put these results in perspective; often the proofs ofproperties of classical examples are simplified. The book will serveas a helpful resource for researchers working in commutative algebra.
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Our Textbook and Related BooksThe text for the class is A First Course in Noncommutative Rings 2nd Edition (Graduate Texts in Mathematics, Book 131), by Tsit-Yuen Lam, Springer (2001). The first edition of this book is available in the ETSU Sherrod Library (QA251.4.L36 1991).Some books related directly to our textbook are: Exercises in Classical Ring Theory, 2nd Edition (Problem Books in Mathematics), T. Y. Lam, NY: Springer (2003). This is a solution manual for all problems in our textbook. This book is available by "online access" through the ETSU Sherrod Library. Search for the book with the online catalog and click on "Online access." You will be prompted for your username and password and then you can view the book. You can also print up to 100 pages or download them in PDF. Lectures on Modules and Rings (Graduate Texts in Mathematics, Book 189) Springer (1999). This is a sequel to A First Course in Noncommutative Rings, also by T.Y. Lam. Exercises in Modules and Rings (Problem Books in Mathematics) by T. Y. Lam, NY: Springer (2007). This is a solution manual to the Lam's Lectures on Modules and Rings.
General BackgroundOn page ix of A First Course in Noncommutative Rings Lam states: "We assume that, from [a standard first-year gradaute course in abstract algebra], the students would have been exposed to tensor products, chain conditions, some module theory, and a certain amount of commutative algebra." Our recent Modern Algebra sequence (MATH 5410 in fall 2017 and MATH 5420 in spring 2018) did not touch on these ring theory topics to any degree. In this class we will primarily rely on the text from the Modern Algebra sequence, Algebra (Graduate Texts in Mathematics 73) by T. W. Hungerford, Springer-Verlag (1974).The most relevant topics are from: Chapter IV: Modules Section IV.1. Modules, Homomorphisms, and Exact Sequences. PDF. (Includes a statement of the "Snake Lemma.")
Supplement. Proofs of Theorems in Section IV.1. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.1. PDF.
Supplement. A Proof of the Snake Lemma. PDF.
IV.5. Tensor Products Section IV.7. Algebras. PDF.
Supplement. Proofs of Theorems in Section IV.7 (partial). PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.7 (partial). PDF.
Chapter IX: The Structure of Rings Section IX.1. Simple and Primitive Rings. PDF.
Supplement. Proofs of Theorems in Section IX.1. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IX.1. PDF.
Section IX.2. The Jacobson Radical. PDF.
Supplement. Proofs of Theorems in Section IX.2. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IX.2. PDF.
Section IX.3. Semisimple Rings. PDF.
Supplement. Proofs of Theorems in Section IX.3 (partial). PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IX.3 (partial). PDF.
IX.4. The Prime Radical; Prime and Semiprime Rings IX.5. Algebras IX.6. Division Algebras General introductory ring theory from Hungerford includes: Chapter III: Rings Section III.1. Rings and Homomorphisms. PDF.
Supplement. Proofs of Theorems in Section III.1. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section III.1. PDF.
Supplement. Quaternions - An Algebraic View. PDF. There is also a PowerPoint version with some history: PowerPoint.
Section III.2. Ideals. PDF.
Supplement. Proofs of Theorems in Section III.2. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section III.2. PDF.
Section III.3. Factorization in Commutative Rings. PDF.
Supplement. Proofs of Theorems in Section III.3. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section III.3. PDF.
Hungerford also addresses commutative ring theory in Chapter VIII (see also the next list of books addressing commutativity): Chapter VIII: Commutative Rings and Modules Section VIII.1. Chain Conditions. PDF.
Supplement. Proofs of Theorems in Section VIII.1. PDF (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section VIII.1. PDF.
VIII.2. Prime and Primary Ideals VIII.3. Primary Decomposition VIII.4. Noetherian Rings and Modules An easier read for some of these introductory ideas is Introduction to Ring Theory (Springer Undergraduate Mathematics Series) by P. M. Cohn, Springer-Verlag (2000). For the table of contents of this book (though no online notes as of August 2018) see my webpage here. 2ff7e9595c
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