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Browse materialsMIT 18.706 · Spring 2023

Noncommutative Algebra

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Roman Bezrukavnikov
Term: Spring 2023
Level: Graduate
What's inside: 6 problem sets · 4 lecture notes · 26 other files · 3 web pages
Materials open: MIT OCW

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Source: MIT OpenCourseWare 18.706 Noncommutative Algebra, Spring 2023, Prof. Roman Bezrukavnikov — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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MIT OpenCourseWare, Prof. Roman Bezrukavnikov, 18.706 Noncommutative Algebra, Spring 2023.

License: CC BY-NC-SA 4.0 · Edition: Spring 2023 · Open the original course on OCW ↗

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Noncommutative Algebra

Before you start

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Study Materials

18.706 (S23), Full Lecture Notes

Lecture 01: Introduction, Simple and Semisimple Modules, Skew Fields

Lecture 02: Semisimple Modules, Socles, Artinian Rings, Wedderburn’s Theorem

Lecture 03: Isotypic Decomposition, Density Theorem, Noetherian and Artinian Properties, Jacobson Radical

Lecture 04: Socle and Cosocle Filtrations, Jacobson Radical, Krull-Schmidt

Lecture 05: Jacobson Radical, Primitive and Semi-Primitive Rings

Lecture 06: Artinian Rings are Noetherian, Projective Covers

Lecture 07: Categories and Morita Equivalence

Lecture 08: Morita Theory Continued: (co)Centers, Functors and Bimodules

Lecture 09: More on Serre Quotients, Abelian Categories

Lecture 10: Exts and Tors, Resolutions

Lecture 11: Koszul Complexes

Lecture 12: Koszul Rings Continued, Bar Complex

Lecture 13: Hochschild (co)Homology Continued, Central Simple Algebras

Lecture 14: Brauer Group and Skolem-Noether Theorems

Lecture 15: Separable Splitting Fields and Cross-Product Algebras

Lecture 16: Cohomological Description of the Brauer Group

Lecture 17: Brauer Groups of Central Simple Algebras, Reduced Norm and Trace

Lecture 18: Azumaya Algebras

Lecture 19: Brauer Group of a Ring Continued, Localization

Lecture 20: Ore Localization, Goldie Theorem

Lecture 21: Goldie Rank and Goldie Theorem

Lecture 22: Goldie Theorem, PI Rings

Lecture 23: Another Proof of Amitsur-Levitski, PI Algebras

Lecture 24: Gelfand-Kirillov Dimension

Lecture 25: Final Class: Noncommutative Geometry

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5

Problem Set 6