Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 250B: Algebra
Approved: 2015-11-01 (revised 2025-05-30, Fuchs and Vazirani)
Units/Lecture:
Every year; 4 units; lecture/term paper or discussion
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
David Dummit, Richard Foote, Abstract Algebra
Prerequisites:
250A or permission via Student Services.
Suggested Schedule:
| Lectures | Sections | Topics/Comments |
|---|---|---|
| 1 | 7.1, 7.2, 7.4 | Lightning review of (undergraduate material) Rings: Ring Homomorphisms, Quotient Rings, Properties of Ideals, Prime ideals. Examples: Polynomial Rings, Matrix Rings,Group Rings |
| 2 | 7.3 | Isomorphism Theorems. (1st, 2nd, 3rd, 4th= Correspondence Theorem) |
| 2 | 7.6, 9.5 | The Chinese Remainder Theorem (9.5 focuses on Z and polynomial rings) |
| 1 | 15.1, 9.6 | Noetherian rings, particularly applied toward proving a PID is a UFD, (optionally Hilbert basis theorem) * |
| 2 | 8.1, 8.2, 8.3, 9.2, 9.3 | Lightning review of (undergraduate material) : Euclidean Domains, Principal Ideal Domains (PIDs), Unique Factorization Domains (UFDs). 9.2, 9.3 focus on the case of polynomial rings. |
| 1 | 7.5, 15.4 | Rings of Fractions, Field of fractions (optionally go deeper into localization of rings in 15.4) |
| 1 | 10.1 | Modules: Basic Definitions and Examples |
| 2 | 10.2 | Quotient Modules, Module Homomorphisms, Isomorphism Theorems |
| 2 | 10.3 | Direct Sums, and Free Modules, universal properties |
| 1 | 10.5, Appendix | Basic category theory, functors |
| 2 | 10.5 | Exact Sequences, Projective modules |
| 2 | 10.4 | Tensor Products of Modules |
| 2 | 11.5, 11.4 | Tensor Algebras. Symmetric and Exterior Algebras, Determinants |
| 1 | 12.1 | Modules over Principal Ideal Domains |
| 2 | 12.2, 12.3 | The Rational Canonical Form, Jordan Canonical Form, Smith normal form |
| 2 | Revisit eigenvalue, characteristic polynomial, Chinese Remainder Theorem | |
Additional Notes:
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page. Many students should have seen basics of commutative rings, ideals, ED, PID, UFD as
an undergraduate.
All of the topics above are easily found in Dummit and Foote, and students may benefit from the ampleness of examples in this book. However, especially for the instructor, it may be better to use smaller and more topic-specific books and notes to get across the big picture for the topics covered. One may choose a more advanced text like Hungerford.
If following Dummit and Foote, note the material in 9.2, 9.3 has substantial overlap with material in Chapter 7, so one might change the sequence above. One may want to cover the material in Chapter 12 before 10.4, 11.4 to leave enough time to these important connections to linear algebra.
* An instructor might want to cut lectures on Ch 11 and focus more on 9.6, 15.1, 15.3, 15.5 material: Polynomials in Several Variables, The Prime Spectrum of a ring, associated primes and primary decomposition, Hilbert's Nullstellensatz over a Field, and Grobner bases.