Lecture No. 1 (Sept. 11.) 0. Introduction, examples. Overview of the literature and possible topics. Basic assumption: all rings will have an identity. (Theorem: Every ring can be embedded as an ideal into a ring with identity.) Examples: endomorphisms of vector spaces, of abelian groups; free algebras; polynomial rings, formal power series ring, ring of formal Laurent series; (semi)group algebras; ring of germs of continuous functions; tensor algebras and important factor algebras; matrix rings and the Peirce decomposition; path algebras and admissible relations.
Lecture No. 2. (Sept. 18.) 1. Primitive rings, density theorem. Primitive rings: definition and inner characterization. The core of a left ideal. k-transitivity and density. The density theorem. Structure of primitive rings. Commutative primitive rings are fields.
Lecture No. 3. (Sept. 25.) 2-transitive implies k-transitive. Prime ideals, prime rings; primitivity and primeness. 2. The Jacobson radical of a ring. Radical of a module, Jacobson radical of a ring. Quasi-invertibility; J(R) as the annihilator of simple modules. J(R) is a two-sided concept. Nil one sided ideals are in the radical.
Lecture No. 4. (Oct. 4.) Primitive ideals; semiprimitivity. Radical of quotients modulo ideals in the radical. The Jacobson radical of a path algebra. The radical of a left artinian ring is nilpotent. The radical of an algebraic algebra is nil. Amitsur's theorem for the nil property of the radical. The radical under various constructions: homomorphism, direct sum. The Jacobson radical of the full matrix ring. The nilradical of a commutative ring. Snapper's theorem about the Jacobson radical of a polynomial ring over a commutative ring. Proof of Snapper's theorem.
Lecture No. 5. (Oct. 9.) The Jacobson radical of a group ring for finite groups; the semiprimitivity problem for group rings over a field of characteristic 0. Superfluous submodules; the Nakayama lemma. Semisimple modules. Characterization of semisimple modules. Semisimple modules with chain conditions. Semisimple rings. The classical Wedderburn–Artin theorem: some characterizations of semisimple rings.
Lecture No. 6. (Oct. 16.) Semisimplicity of rings is a two-sided concept. Theorem of Hopkins: Artinian rings are Noetherian. Theorem of Levitzki: in a left noetherian ring every nil one-sided ideal is nilpotent. 3. Structure theory, commutativity theorems. Wedderburn's theorem: every finite division ring is commutative. Finite multiplicative subgroups of division rings of finite charcteristic. Jacobson's theorem: if in R for every element a there is an exponent n(a)>1 for which an(a)=a then the ring is commutative. Key lemma about division rings.
No lecture (statutory holiday) (Oct. 23.)
Lecture No. 7. (Nov. 6.) Finishing the proof of Jacobson's theorem. Some other commutativity results (without proof). 4. Direct sum decomposition of modules and chain conditions. Inner characterization of a direct sum.
Lecture No. 8. (Nov. 13.) Local rings. Characterization of local rings. Modules with local endomorphism ring are (strongly) indecomposable. Azumaya's theorem about the uniqueness of direct sum decompositions (without proof). Indecomposable modules of finite composition length are strongly indecomposable. The classical Krull–Schmidt theorem. Rings of finite representation type; Auslander's theorem (without proof). Injective modules.
Lecture No. 9. (Nov. 20.) The injective test lemma. Injective abelian groups, divisibility. Characterization of injective modules: they are direct summands whenever they appear as a submodule. Essential submodules. Injective envelope. The minimality and uniqueness of an injective envelope. The existence of an injective envelope. Indecomposable injective modules are strongly indecomposable. Characterization of indecomposable injective modules over left artinian rings. The following are equivalent for a ring: the ring is noetherian; direct sums of injective modules are necessarily injective. – Statements to be read from the course notes: Further equivalent characterizations of the noetherian property: every indecomposable injective module is the direct sum of indecomposable injectives. A ring is artinian if and only if every injective module is the direct sum of injective envelopes of simple modules.
Lecture No. 10. (Nov. 27.) Characterizations of the noetherian property (ccontinued, mostly without proofs. Example of a module having two inequivalent indecomposable decompositions. 5. Categories and functors. The Hom and tensor functors. Basic definitions. Examples of categories. Category of chain complexes; the homotopy category of chain complexes.
Lecture No. 11. (Dec. 4) Functors, covariant and contarvariant, the Hom functors. Preadditive categories, additive functors. Natural transformations and natural isomorphisms. The Hom functors. Module structure on Hom(M,N) in case M is a bimodule. Equivalence of categories. Example of a categorical equivalence which is not an isomorphism. Notions of a monomorphism, epimorphism and isomorphism. Products and coproducts in categories, specific examples. The product of injectives (the coproduct of projectives) is always injective (projective). R-balanced maps and tensor product of modules.
Lecture No. 12. (Dec. 5.) Construction of the tensor product of modules. Module structure on M⊗N if M is a bimodule. Tensor product as a bifunctor. Natural isomorphisms involving the tensor product and Hom functors. Adjoint pairs of functors: the adjointness of the Hom and tensor funcotrs (not in full detail). Exact and half exact functors. Examples. Injective, projective and flat modules. Injective, projective and flat modules. Every module can be embedded into an injective module.