This webpage gives a list of books and research papers which may be useful in the fall 2018 Noncommutative Ring Theory (MATH 5900) class. The references are borken into general classes.

1. 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.
2. 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.
3. 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.

• 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

• 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.
  Commutative Algebra On page 2 of A First Course in Noncommutative Rings Lam comments "...since much of our work is an attempt to extend results from the commutative setting to the general setting, it will pay handsomely if the reader already has a good idea of what goes on in the commutative case." He specifically mentions the following three books on the commutative case.
  1. Commutative Algebra 1 (Graduate Texts in Mathematics 28), O. Zariski and P. Samuel, Springer-Verlag (1975) and Commutative Algebra 2 (Graduate Texts in Mathematics 29), O. Zariski and P. Samuel, Springer-Verlag (1976). This volumous work (750 pages total) is available at the ETSU Sherrod Library (QA251.Z3).
  2. Introduction to Commutative Algebra, M. F. Atiyah and I. G. Macdonald, Addison-Wesley Publishing Co. (1969). This brief work (less than 130 pages) appears to be a readable, good reference for the topic. This is available at the ETSU Sherrod Library both in print (QA251.A8) and through online access.
  3. Commutative Rings (Revised Edition), I. Kaplansky, University of Chicago Press (1974). This book is also short (170+ pages) covering prime ideals, Noetherian rings, Macauly rings, regular rings, and homological aspects of ring theory. My impression is that this is a harder read than Atiyah and Macdonald's work. The first edition is available through the ETSU Sherrod Library (QA247.K3 1970)
  Sets of Roots of Quaternionic Polynomials
  1. S. De Leo, G. Ducati, V. Leonardi, Zeros of Unilateral Quaternionic Polynomials, Electronic Journal of Linear Algebra 15, Article 25 (2006).
  2. S. Eilenberg and I. Niven, The "Fundamental Theorem of Algebra" for Quaternions, Bulletin of the American Mathematical Society 50, 246-248 (1944).
  3. G. Gentili and C. Stoppato, Zeros of Regular Functions and Polynomials of a Quaternionic Variable, Michigan Math. J. 56, 655-667 (2008).
  4. B. Gordon and T. Motzkin, On the Zeros of Polynomials over Division Rings, Transactions of the American Mathematical Society 116, 218-226 (1965). See also: B. Gordon and T. Motzkin, Correction to the Paper On the Zeros of Polynomials over a Division Ring, Transactions of the American Mathematical Society 122(2), 547 (1966).
  5. I. Nivens, Equations in Quaternions, The American Amathematical Monthly 48(10), 654-661 (1941).
  6. A. Pogorui and M. Shapiro, On the Structure of the Set of Zeros of Quaternionic Polynomials, Complex Variables 49(6), 379-389 (2004).
  An Analytic Theory of Functions of a Quaternionic Variable
  1. G. Gentili and D. Struppa, A new Theory of Regular Functions of a Quaternionic Variable, Advances in Mathematics 216, 279-301 (2007). [This is a paper of central importance! The author's show that regular functions of a quaternionic variable have a power series with radius of convergence and that the Maximum Modulus Theorem, Liouville's Theorem, Morera's Theorem, and a version of Schwarz's Theorem all hold. They explore integration and state a Cauchy Integral Formula. This paper leads me to believe that many of the Enestrom-Kakeya Type Theorems can be carried over to the quaternionic variable setting. For such results in the complex setting, see my paper with N. K. Govil, The Enestrom-Kakeya Theorem and Some of Its Generalizations.]
  Complex-Type Results in the Quaternionic Setting
  1. S. G. Gal and I. Sabadini, On Bernstein and Erdos-Lax's Inequalities for Quaternionic Polynomials, C. R. Acad. Sci. Paris, Ser. I 353, 5-9 (2015). [The authors give a version of the Erdos-Lax Theorem; it does not carry over to this setting from the complex setting without some modification.]
  2. R. Ghiloni and A. Perotti, The Quaternionic Gauss-Lucas Theorem, arXiv: 1712.00744v2 (March 2018).
  3. F. Vlacci, The Gauss-Lucas Theorem for Regular Quaternionic Polynomials, in Hypercomplex Analysis and Applications I. Sabadini and F. Sommen eds., Basel: Springer, 275-282 (2011).
  4. F. Vlacci, Vieta's Formulae for Regular Polynomials of a Quaternionic Variable, Adv. Geom. 17(4), 435-440 (2017).
  5. Z. Xu, The Bernstein Inequality for Slice Regular Polynomials, arXiv: 1602.08545v1 (February 2016). [This includes an Ankeny-Rivlen type result.]
  Computational Results Concerning Zeros of Quaternionic Functions
  1. A. Damiano, G. Gentili, D. Struppa, Computations in the Ring of Quaternionic Polynomials, Journal of Symbolic Computation 45, 38-45 (2010).
  2. D. Janovska and G. Opfer, A Note on the Computation of all Zeros of Simple Quaternionic Polynomials, SIAM J. Numer. Anal. 48(1) 244-256 (2010).
  The Octonions
  1. J. C. Baez, The Octonions, Bulletin of the American Mathematical Society 39(2), 145-205 (2002).
  2. J. C. Baez, Errate for "The Octonions," Bulletin of the American Mathematical Society 42(2), 213 (2005).
  3. R. Ghiloni and A. Perotti, Zeros of Regular Functions of Quaternionic and Octonionic Variable: A Division Lemma and the Camshift Effect, Annali di Matematica 190, 539-551 (2011).
  4. R. Ghiloni and A. Perotti, A New Approach to Slice Regularity on Real Algebra, in Hypercomplex Analysis and Applications, I. Sabadini and F. Sommen eds., Basel: Springer, 109-123 (2011). A copy is online here.
  5. J. Liao and X. Li, On k-Analytic Functions of an Octonionic Variable, Advances in Applied Clifford Algebras 27, 3149-3166 (2017).

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  Last updated: December 5, 2018.