NONCOMMUTATIVE RING THEORY - Fall 2018

Emmy Noether, 1882-1935

Joseph H. M. Wedderburn, 1882-1948

Emil Artin, 1898-1962

Nathan Jacobson, 1910-1999


Dr. Keaton lecturing on August 28.

COURSE: MATH 5090-001

TIME: 2:45-3:40, PLACE: Gilbreath Hall 304

INSTRUCTORS: Robert "Dr. Bob" Gardner and Rodney Keaton, OFFICE: Room 308F of Gilbreath Hall (Gardner)

OFFICE HOURS: TR 3:35-4:00 and by appointment (Gardner) PHONE: 439-6979 (Gardner), Math Office 439-4349

E-MAIL: gardnerr@etsu.edu
WEBPAGE: http://faculty.etsu.edu/gardnerr/gardner.htm

TEXT: 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).

A solution manual for all problems in this book is available in Exercises in Classical Ring Theory, 2nd Edition (Problem Books in Mathematics), T. Y. Lam, NY: Springer (2003). 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. A sequel to A First Course in Noncommutative Rings is Lam's Lectures on Modules and Rings (Graduate Texts in Mathematics, Book 189) Springer (1999). A solution manual to this book is Exercises in Modules and Rings (Problem Books in Mathematics) by Lam, NY: Springer (2007). We will also use some research papers which address polynomials and regular functions of a quaternionic variable. A list of such papers and additional references is available online:
Noncommutative Ring Theory References.

ABOUT THE COURSE: As stated on the proposal for this class (as prepared by Ms. Powers and Mr. Carney): "The highlights of the subject matter of this course are: basic ring theory and terminology, Wedderburn-Artin theory of semisimple rings, Jacobson’s theory of the radical, representation theory of groups and algebras, prime and semiprime rings, primitive and semiprimitive rings, revision rings, ordered rings, local and semilocal rings, and perfect and semiperfect rings. Of course, the entirety of the subject matter of this course is not limited to these subtopics." This is an ambitious list of topics and will likely be greatly shortened. We will explore some recent research results concerning the structure of the set of roots of polynomials of a quaternionic variable and the extension of some polynomial results from the complex setting to the quaternionic setting.

GRADING: As stated in the proposal for this course (as written by Mr. Carney and Ms. Powers), students will "develop proficiency in the material covered by the text by working through the book and its exercises." Students will type up notes and exercise solutions that could be used in a future course in noncommutative rings. The instructor will evaluate this material and assign numerical grades. The final grade will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate. Based on the assignment of grade points by ETSU, the plus and minus grades should be given on a 3 point subscale. For example, a B+ corresponds to an average of 87, 88, or 89; an A- corresponds to an average of 90, 91, or 92; an A corresponds to an average of 93 to 100 (ETSU does not grant A+ grades, and the lowest passing grade for a graduate course is C), etc.

ONLINE NOTES: As material is developed during the class, it will be prepared in LaTeX and posted online in PDF. This material will include definitions, some motivational comments, and statements of lemmas, theorems, and corollaries, and detailed proofs. The is will be posted at:

http://faculty.etsu.edu/gardnerr/5900/notes-noncommutative-rings.htm.

OUTLINE OF TOPICS:
We will attempt to cover topics from this list:
Chapter 1: Wedderburn-Artin Theory.
Definitions, examples, exercises, semisimple rings.
Chapter 2: Jacobson Radical Theory.
Definitions, examples, exercises, Jacobson radical, group rings.
Chapter 3: Introduction to Representation Theory.
Definitions, examples, exercises, modules, representations of groups, linear groups.
Chapter 4: Prime and Primitive Rings (optional).
Definitions, examples, exercises, prime radical, prime and semiprime rings, the Density Theorem, subdirect products.
Chapter 5: Introduction to Division Rings.
Definitions, examples, exercises, tensor products, polynomials and the structure of their zero sets.
Chapter 6: Ordered Structures in Rings.
Definitions, examples, exercises, preorderings and orderings, ordered division rings.
Chapter 7: Local Rings, Semilocal Rings, and Idempotents (optional).
Definitions, examples, exercises.
Chapter 8: Perfect and Semiperfect Rings (optional).
Definitions, examples, exercises. The Structure of Zero Sets of Polynomials of a Quaternionic Variable.

OTHER RESOURCES. The following were mentioned in class:

  1. To access the Mathematical Reviews: Go to the Sherrod Library online catalog. Search for "MathSciNet" and choose "view online." You can then search for any publication which is in the Mathematical Reviews database of the American Mathematical Society. In many cases, this will give you access to PDF versions of the papers which you can view and download. MathSciNet also allows you to search author or keyword. This is a super-useful research tool!
  2. Modern Algebra Supplement: The Quaternions: An Algebraic Approach.
  3. Modern Algebra Supplement: The Quaternions: An Algebraic and Analytic Exploration (presented at Auburn University, August 7, 2018.

IMPORTANT DATES: (see http://www.etsu.edu/etsu/academicdates.aspx for the official ETSU calendar; accessed 5/16/2018):

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Last updated: September 30, 2018.