## Basic course description

Math 200b is the second quarter of UCSD's three-quarter graduate-level abstract algebra course. Math 200a is a prerequisite. Please come talk to me if you believe this is the right course for you although you have not taken Math 200a. The main aim of the course is to give PhD and masters students in mathematics sufficient background for their further studies, and to prepare these students for the qualifying exam in algebra given in May 2020 and again in September 2020. The text will be Dummit and Foote, "Abstract Algebra", 3rd edition.

### TA and Professor Contact Information

Professor Rogalski: e-mail drogalski@ucsd.edu.

• Lecture: MWF 12pm-12:50pm, 5402 AP&M
• Office hours: W 1-2pm, Th 4-5pm in 5131 AP&M

TA: Jake Postema: e-mail jpostema@ucsd.edu

• Office hours: Tu 2-3pm, W 4-5pm in 5801 AP&M

### Schedule of lectures

We will cover much of Chapters 10 and 12-14 of the text, but not always in the order the topics are presented in the text. The following schedule will be updated as we go with information about what we covered on particular days.
• 1/6/2020 (Section 10.1-10.2) Definition of a left R-module. Examples. Homomorphisms, submodules, factor modules, and examples. Fundamental homomorphism theorems for modules. Left R-modules are the same as right R^op modules. The set Hom_R(M,N) of module homorphisms from M to N is an Abelian group and is an R-module if R is commutative.
• 1/8/2020 (Section 10.1-10.2) End_R(M) = Hom_R(M,M) is a ring where the multiplication operation is composition, called the endomorphism ring of the module M. Examples: if V is an n dimensional vector space over F then End_F(V) is isomorphic to M_n(F). If R is a left R-module by multiplication then End_R(R) is isomorphic to R^op. An R-module structure on an Abelian group M is the same thing as a ring homomorphism R --> End_Z(M). An F[x]-module where F is a field is the same as an F-vector space V together with a choice of F-linear map V --> V which tells you how x acts. Definition of the direct sum and product of an indexed set of R-modules.
• 1/10/2020 (Section 10.3) The submodule generated by a subset. Finitely generated modules. Cyclic modules. Every cyclic left R-module is of the form R/I for a left ideal I. Free modules. The universal property of a free module. A direct sum of copies of R as a left module is free, and every free module is of this form. Any two free modules with bases of the same cardinality are isomorphic.
• 1/13/2020 (Section 12.1) Internal direct sums. Indecomposable modules. Examples. Rank of a module over an integral domain R. There are at most n R-linearly independent elements in R^n; in particular R^n has rank n. Statement of the main theorem classifying finitely generated modules over a PID (invariant factor form). Examples where it fails for infinitely generated modules or for finitely generated modules over non-PIDs. Torsion submodule of a module over an integral domain. Comments on how to prove the uniqueness part of the classification theorem.
• 1/15/2020 (Section 12.1) Proposition: If R is a PID, a submodule N of R^n is also free, and one can choose a basis y_1, ..., y_n of R^n such that a_1y_1, ..., a_ny_n is a basis of N, where each a_i divides a_{i+1}. Proof of the main theorem classifying f.g. modules over R using this proposition.
• 1/17/2020 (Section 12.1) The uniqueness part of the classification theorem of f.g. modules over PIDs (invariant factor form). The Elementary divisor form of the theorem. Examples of calculating the elementary divisors from the invariant factors and vice versa.
• 1/20/2020 MLK day (no class)
• 2/10/2020 Midterm Exam (in class)
• 3/18/2020 Final Exam 11:30am-2:30pm