Lecture topics

Math 6250: Theory of Rings, Spring 2012

Lecture Topics

Date	What we discussed/How we spent our time
Jan 18	This will be a course on noncommutative ring theory. Grades will be based on HW. We introduced rings as algebraic models of End(A), where A is an abelian group.
Jan 20	We defined representations of rings, k-algebras, and representations of k-algebras.
Jan 23	Categories and their representations.
Jan 25	Semisimple modules and rings. We proved that a complemented modular lattice is relatively complemented. [This is part of a proof that a module is a (direct) sum of simple modules iff its submodule lattice is complemented.]
Jan 27	A module is a sum of simple submodules iff it is a direct sum of simple submodules. Any module with a complemented submodule lattice is a sum of simple submodules.
Jan 30	A module that is a sum of simple submodules has a complemented submodule lattice. A ring R is semisimple iff Sub(_RR) is complemented. A semisimple ring is left artinian, left noetherian, and has only finitely many isomorphism types of simple modules. If D is a division ring, then M_n(D) is semisimple. In fact, a finite product of rings of this type, M_n1(D1)×M_n2(D2)×…×M_nk(Dk), is semisimple.
Feb 1	Lm. R≅End(_RR) Schur's Lm. If S is a simple R-module, then End(S) is a division ring. Schur's Lm for algebras. Let k be a field, A a k-algebra, S a simple A-module. (i) D:=End(S) is a k-division algebra. (ii) If A is finite dimensional over k, then so are S and D. (iii) If k is algebraically closed and A is f.d. over k, then D=k.
Feb 3	Snow day.
Feb 6	Matrix representation of homomorphisms between direct sums. Wedderburn-Artin Theorem.
Feb 8	We proved a theorem characterizing simple artinian rings in several ways. We discussed equivalence of categories and started a proof that _RMod is equivalent to _{M_n(R)}Mod.
Feb 10	We finished the proof that _RMod is equivalent to _{M_n(R)}Mod.
Feb 13	Jacobson radical: basic properties.
Feb 15	The radical acts nilpotently on modules of finite length. The radical of a left artinian ring is nilpotent. A left artinian ring is J-ss iff it is ss.
Feb 17	Thm. (Hop.-Lev.) Any left artinian ring is left noetherian.
Feb 20	Thm. (Hop.-Lev.) TFAE: (i) _RR has finite length. (ii) R is left artinian. (iii) (a) R/J is ss, (b) J is nilpotent, (c) Each power J^k is f.g. as a left ideal. Thm. (Hop.-Lev., module version) If R is semiprimary and M is an R-module, then TFAE: (i) M is artinian. (ii) M is noetherian. (iii) M has a composition series. Thm. U(R) is a union of J-cosets, and U(R)/J = U(R/J). Thm. (Amitsur) If k is a field and A is a k-algebra of small dimension (dim_k(A)<\|k\|), then rad(A) is nil.
Feb 22	Behavior of the radical with respect to quotients.
Feb 24	The radical of a polynomial ring, part 1.
Feb 27	The radical of a polynomial ring, part 2. A discussion of Koethe's conjecture.
Feb 29	Groups rings and algebras. We started the proof of the strong form of Maschke's Theorem.
Mar 7	We finished the proof of Maschke's Theorem.
Mar 9	We discussed two open questions: Which groups algebras over fields are J-s.s.? How do you compute the radical of a group algebra over a field? We began discussing representation theory.
Mar 12	Terminology: representation of a k-algebra, irreducible representation, reducible representation, completely reducible representation, module affording a representation, character of a representation, representation affording a character. Thm. If A is a f.d. k-algebra, A/rad(A) ≅ M_n₁(D₁)×…× M_{n_r}(D_r), and M₁, …, M_r are the types of simple A-modules, then (i) dim_k(M_i) = n_idim_k(D_i) (ii) dim_k(A) = dim_k(rad(A)) + ∑ n_i²dim_k(D_i) (iii) the natural map A→M_{n_i}(D_i) is surjective For A = M₂(ℝ), we computed the characters of ℝ² and A as A-modules. Then we did the same thing with A replaced by the ring of upper triangular 2×2 matrices over ℝ. Thm. If 0 → L→ M→ N→ 0 is exact, then χ_M = χ_L + χ_N. Cor. χ_M is the sum of the characters of the composition factors of M (counting multiplicity).
Mar 14	Thm. If k is a field of characteristic zero, A is a f.d. k-algebra, and M is a f.g. A-module, then χ_M determines the composition factors of M. (In fact, the proof shows that the same result holds in positive characteristic provided char(k) > dim_k(M).) We gave examples to show that a restriction on characteristic is essential in the above theorem, namely the case where k is a finite field, or when A is an inseparable field extension of k. We discussed χ_K in the situation where A=K is a field extension of k.
Mar 16	We began discussing representations of finite groups. We described the f.d. representations of G=ℤ₅ over k=ℚ, ℝ, and ℂ, along with the associated characters.
Mar 19	We began discussing representations of finite groups over ℂ. We calculated the character table of S₃. We started on these notes, and explained items (1)-(8).
Mar 21	We explained how representations and characters of G/N can be lifted to G. We used this to calculate the character table of D₄. We discussed permutation representations and their characters. We explained items (14)-(16) of the character theory notes.
Mar 23	♦ ℂ[G] ≅ M_n₁(D₁)× …×M_{n_r}(D_r), D₁=…=D_r=ℂ ♦ \|G\|=∑ n_i² ♦ χ_reg = ∑ n_i χ_i ♦ r = dim_ℂ(Z(ℂ[G])) = class number of G We found the character tables of ℤ₃ and the alternating group A₄. (We found the 3-dimensional irreducible character of A₄ both arithmetically and geometrically.) We explained items (17) and (18) of the notes.
Apr 2	If U and V are G-modules, then we explained how to make U^* and Hom_ℂ(U,V) into G-modules. We proved that ♦ χ_U^*(g) = χ_U(g^-1) (= χ_U(g)). ♦ χ_{Hom_ℂ(U,V)}= χ_U χ_V.
Apr 4	We proved that dim_ℂ(V^G)=(1/\|G\|)*∑ χ_V(g) We defined the inner product of class functions and showed that ⟨χ_U,χ_V⟩ = dim_ℂ(Hom_ℂ[G](U,V))
Apr 6	We proved row and column orthogonality. We observed that χ is irreducible iff ⟨χ,χ⟩ = 1. We computed the character table of S₄.
Apr 9	We proved not-Burnside's Lemma. We discussed internal vs. external tensor product. The main points of the discussion were: ♦ Representations/characters of internal tensor products are derived from those of external tensor products via the diagonal map Δ:G→G×G. ♦ The permutation representation of G×H on X×Y is the external tensor product of the permutation representation of G on X with the permutation representation of H on Y. ♦ χ_U#V(g,h) = χ_U(g)χ_V(h). ♦ ⟨χ_U#V,χ_M#N⟩= ⟨ χ_U, χ_M ⟩ ⟨ χ_V, χ_N ⟩. ♦ The irreducible representations/characters of G×H are exactly the #'s of the irreducible characters of G with those of H.
Apr 11	We proved items (9)-(13), (27) of the notes.
Apr 13	We proved that a complex number α satisfies a monic integer polynomial iff it lies in (equivalently, generates) a subring of ℂ that is finitely generated as an abelian group. We took these equivalent conditions for the definition of algebraic integer. We showed that the algebraic integers, 𝔸, form a ring, and that 𝔸∩ℚ = ℤ. We showed that character values are algebraic integers. We also showed that if K_j is the jth conjugacy class of G, k_j = \|K_j\|, g_j∈K_j is an element representing this class, κ_j is the sum of the elements of the class, χ_i is an irreducible character afforded by a representation ρ_i:ℂ[G]→End(V), then ρ_i(κ_j) = (k_jχ_i(g_j)/χ_i(1))I.
Apr 16	We finished the character theory handout. (Finally!)
Apr 18	We constructed the character table of A₅. We proved Burnside's theorem that no simple group has a (nonidentity) conjugacy class of prime-power order. We proved Burnside's theorem that any group whose order is divisible by at most two distinct primes is solvable.
Apr 20	We discussed the group determinant. We identified groups that act on the character table of G. (The group of linear characters, which is isomorphic to G/[G,G], acts on all irreducible characters. Consequence for the symmetric group: odd degree characters come in pairs.)
Apr 23	If \|G\|=n and ω is a primitive nth root of unity, then Gal(ℚ[ω]/ℚ)≅ℤ_n^* acts on the character values. This has the effect of permuting the column headings and the row headings of the character table, X. This yields uniquely determined permutation matrices P_k and Q_k such that, if σ_k∈Gal(ℚ[ω]/ℚ), then P_kX=σ_k(X)=XQ_k. The maps k↦P_k and k↦Q_k are isomorphic permutation representations of Gal(ℚ[ω]/ℚ). This leads to the definitions of rational and real elements/characters/groups. S_n is rational (hence has rational integral character values) and D_n is real.
Apr 25	Jacobson Density Theorem. The usual setup is that R is a ring, M is a left R-module, E= End(_RM), S = End(M_E), and φ:R→S:r↦λ_r is the homomorphism of R into its double centralizer S. Goals: (i) to show that when M is semisimple, then φ maps R onto an M-dense subring of S, and (ii) to show that if M is a direct sum of the simple left R-modules (one copy of each type), then S is isomorphic to a product of endomorphism rings of vectors spaces over division rings. Together these show that for any ring R there is a dense embedding of R/rad(R) into a product of endomorphism rings of vector spaces.
Apr 27	We achieved the goals identified in the previous lecture. We defined primitive and semiprimitive rings. We explained a proof strategy for statements about rings: prove them for primitive rings, then semiprimitive rings, then lift modulo the radical.
Apr 30	Every algebraic structure is a subdirect product of subdirectly irreducible (SI) quotients. An identity holds in an algebraic structure iff it holds in every SI quotient. Jacobson's Commutativity Theorem: If a ring R satisfies xⁿ=x for some n, then it satisfies xy=yx. If there is a counterexample to the therem, then it can be taken to be SI. Any SI counterexample is a division ring whose maximal subfields have size at most n. It remains to show (i) if D is a division ring with a finite maximal subfield, then D is finite, and (ii) finite division rings are commutative.
May 2	We proved part of the double centralizer theorem.
May 4	We finished the proof of the double centralizer theorem. We proved Wedderburn's Little Theorem. We finished the proof of Jacobson's Commutativity Theorem.