Lecture:

Spletna Učilnica

Description

This course is an introduction to noncommutative algebra.

Contents

Week 1: Basics, Examples

Notes 2.10. Recalling rings, algebras, modules. Basic examples of noncommutative rings.

Reading: [Lam01, §1], [FD93, §0].

Week 2: Noetherian, Artinian, Composition Series

Notes 9.10. Noetherian and artinian modules and rings. Simple modules. Schur's Lemma. Composition Series. Zassenhaus Lemma.

Reading: parts of [FD93, §0 and §1], [Lam01, §1], [Bre25, Chapter 3]

Week 3: Semisimple Modules and Rings.

Notes 16.10. Jordan-Hölder Theorem and modules of finite length. Semisimple modules, endomorphism rings of semisimple modules, semisimple rings, Wedderburn-Artin Theorem formulated (proof next week).

Reading: [FD93, §1], [Lam01, §2 and §3], [Bre25, Sections 3.8-3.9]. For an alternative nice and elementary approach to a slightly weaker version of Wedderburn's Structure Theorem, see [Bre25, Chapter 2], in particular [Bre25, Section 2.9]

Week 4: Simple Artinian Rings, Maschke's Theorem, Jacobson Radical

Notes 23.10. Proof of Wedderburn-Artin; Characterization of simple artinian rings; Maschke's Theorem on semisimple group algebras; Definition and basic properties of the Jacobson radical.

Reading: [FD93, §1 and §2] (beware Remark 1 on p.58 contains mistakes); [Lam01, §3, §4]; [Bre25, Sections 3.9, parts of Ch. 5 for the Jacobson radical] (the structure and order of results in this book is different though).

Week 5: Jacobson Radical, Nakayama's Lemma

Notes 30.10. Jacobson radical, Hopkins-Levitzki, another characterization of semisimple rings, Nakayama's Lemma, primitive rings.

Reading: [FD93, §2]; [Lam01, §4, §11].

Week 6: Jacobson Density Theorem, Primitive Rings

Notes 6.11. Bimodules, Jacobson Density Theorem, Structure Theorem for Primitive Rings; Jacobson's Commutativity Theorem.

Reading: [FD93, §5], [Lam01, §11, §12], [Bre25, Ch. 5].

Week 7: Herstein's Lemma, Wedderburn's Little Theorem, Tensor Products of Modules

Notes 13.11. Proof of Herstein's Lemma and Wedderburn's Little Theorem (finite division rings are fields). Discussion of tensor products of modules and bimodules over nc rings (tensor-hom adjunction).

For Tensor Products over Fields (this is all we need later on): [Bre25, 5.1-5.3]. Herstein's Lemma and Wedderburn's Little Theorem have (shorter) proofs using a bit more theory (e.g. [Lam01, §13], [Bre25, §1.8], [FD93, Thm. 3.18]). The proof of Herstein's Lemma from the lecture can be found in §3.1 of Herstein's "Noncommutative Rings" (1971), the one of Wedderburn's Little Theorem in [Lam01, §13]. Tensor Products of Modules over nc rings are treated in various standard algebra texts, e.g., Hungerford's "Algebra", Chapter IV.5.

Week 8: Tensor Products of Algebras

Notes 20.11. Tensor products of algebras, central algebras, simple algebras. Up to and including proof of Lemma 6.5. Proof Thm. 6.4 next week.

Reading: [FD93, §3], [Bre25, §4.4-4.8].

Week 9: Separable Algebras, Skolem-Noether, Centralizer Theorem

Notes 27.11. Finish tensor products of simple algebras, separable algebras, Skolem-Noether Theorem, Centralizer and Double Centralizer Theorem.

Reading: [FD93, §3], [Bre25, §4.8-4.10]

Week 10: Splitting Fields, Frobenius Theorem, Brauer Group

Notes 4.12. Splitting Fields, Separable Splitting Fields, Frobenius Theorem, Definition of the Brauer Group.

There was a question about which groups appear as Brauer groups ("Inverse Problem for Brauer group"), see MathOverflow.

Reading: [FD93, §4], [Bre25, §4.11].

Week 11: Brauer Group, Crossed Products

Notes 11.12. Brauer group of \(\mathbb{Q}\) (without proofs), Relative Brauer Groups and Crossed Product Algebras. (Up to Theorem 7.12, the proof of which will follow next week).

Reading: [FD93, §4].

Week 12: Brauer Group and Cohomology

Notes 18.12. Bijection between Brauer group and 2-cocycles modulo equivalence; group cohomology; isomorphism between Brauer group and $H^2$, Brauer group is torsion.

Reading: [FD93, §4].

Week 13: Hilbert's Theorem 90, Direct-Sum Decompositions of Modules

Notes 8.1. Hilbert-Noether Theorem on the first Galois cohomology group, Hilbert's Theorem 90; Indecomposable Modules, Existence of Decomposition into Indecomposables, Local Rings

Reading: [FD93, §4], [Lam01, §19].

Week 14: Krull-Schmidt-Remak-Azumaya

Notes 15.1. KRSA for a) modules of finite length b) f.g. modules over module-finite algebras over complete local noetherian commutative rings.

Reading: [Lam01, §21].

Bibliography

Main References

The course is mostly based on a combination of the following books.

[Bre25]

M. Brešar. Introduction to Noncommutative Algebra. 2nd edition, Springer, 2025.

[FD93]

B. Farb, R. K. Dennis. Noncommutative Algebra. GTM 144. Springer, 1993.

[Lam01]

T.Y. Lam. A First Course in Noncommutative Rings. 2nd edition, Springer, 2001.

Additional Resources

[Alu09]

P. Aluffi. Algebra: Chapter 0. GTM 104. American Mathematical Society, 2009.

[Hun80]

T.W. Hungerford. Algebra. (Reprint of the 1974 original) GTM 73. Springer, 1980.

[Lam99]

T.Y. Lam. Lectures on Modules and Rings. GTM 189. Springer, 1999.

[Row08]

L.H. Rowen. Graduate Algebra: Noncommutative View. Graduate Studies in Mathematics 91. American Mathematical Society, 2008.