Description
We will cover material related to rings and modules. Some other related material (e.g. commutative algebra, category theory or homological algebra) will be covered if time permits. Prerequisites: Undergraduate level abstract algebra (such as MATH0430) Text: Dummit and Foote ''Abstract Algebra'' Other textbook: Knapp's Basic Algebra |
Announcements
o Dec. 8: Final exam: It is 2 hours long. There will be questions to state definitions and theorems (without proof). Most problems will be from or similar to suggested homework problems. It covers 7.1-7.6, 8.1-8.3, 9.1-9.6 (only up to p. 323, that is, not including material about Buchberger's algorithm and criterion), 10.1-10.5, 12.1-12.3 Somethings you need to know (review): - Group ring RG, ring of integers in a quadratic field Q(sqrt{d})(quadratic integer ring) - Sum, intersection, product of ideals, prime ideal, maximal ideal, quotient by prime is an integral domain, quotient by maximal ideal is a field, first isomorphism theorem, radical of an ideal - Definition of variety V of an ideal I in a polynomial ring F[x_1, ..., x_n] (= zero set of polynomials in I) and coordinate ring of V (=F[x_1, ..., x_n] / I). - Rings of fraction or in other words localization of an integral domain with respect to a multiplicative set (for example complement of a prime ideal), local ring (has unique maximal ideal), field of fractions of an integral domain, relationship between prime ideals in R and in its localization S^-1R. - Field of fractions of the power series ring F[[x]] is the Laurent series field F((x)). - Chinese Remainder Theorem and its examples when R=Z or a polynomial ring F[x]. - Euclidean domain, PID and UFD, prime element, irreducible element, gcd in PID and UFD, in UFD prime=irreducible. - Examples: Z[i] is ED, F[x] is PID, F[x,y] is not PID but is UDF Z[sqrt{-5}] is not UFD and hence not PID nor ED. Example of an ideal in Z[sqrt{-5}] that is not principal. - Examples of PID: Z, F[x], ring of formal power series F[[x]], ring of p-adic integers Z_p - Prime/irreducible elements in ring of Gaussian integers Z[i], unit elements in Z[i] - Noetherian ring, equivalent definitions: (1) every ideal is f.g. (2) every increasing chain of ideals stops, proof of equivalence of (1) and (2). - Gauss's lemma, R is UFD iff R[x] is UFD - Irreducibility criteria for polynomials e.g. Eisenstein criterion for irreducibility in Z[x] and Q[x], cyclotomic polynomial \phi_p(x) for a prime number p and proof that it is irreducible. - Hilbert basis theorem - Definition of a term order (or monomial order) on a polynomial ring (in several variables), definition of LT(f), that is, leading term of a polynomial f (in several variables), generalized division algorithm (that is, dividing a polynomial f by other polynomials g_1, ..., g_r), definition of Grobner basis (for an ideal), theorem that the remainder in generalized division is unique if we have a Grobner basis. - Basic definitions regarding modules such as submodule, quotient module, module homomorphism, direct sum, Hom(M, N) of modules M and N. - An F[x]-module is equivalent to a vector space V over F with a linear transformation A:V to V. - An ideal I in R means I is an R-submodule of R (regarded as an R-module). - Free module, rank of a free module (theorem that it is well-defined), universal property of free module (over a set A). - Finitely generated module, torsion submodule (when the ring R is integral domain). - Noetherian module (when R is commutative), equivalent conditions of Noetherian (every increasing chain of submodules stops, every submodule is finitely generated), a module if finitely generated iff it is quotient of a free module of finite rank. - Tensor product of modules, universal property of tensor product (in commutative and non-commutative cases), some examples of tensor product (in Section 10.4). - For an R-module M we have M \otimes R is isomorphic to M. - Tensor product and ring of fractions (Exercise 8 in Sec. 10.4) - For vector spaces V, W, the vector space Hom(V, W) is naturally isomorphic to V^* \otimes W - Definitions of: complex of modules, exact sequence, short exact sequence, split short exact sequence, equivalent conditions that a short exact sequence is split - Definition of a (covariant or contravariant) functor (between categories) and definition of an exact functor. - Taking Hom is left exact, tensor product is right exact. - Universal properties of projective and injective modules and equivalent conditions defining them - Examples and non-examples of projective and injective modules, projective and injective modules when R=Z - Definition of flat module, some examples of flat modules. - Fundamental Theorem of f.g. modules over a PID: both versions (invariant factors, elementary divisors). - Smith Normal Form (statement of theorem) - Example of Smith Normal Form: submodules in Z^n, example of a problem: given a submodule N in M=Z^n find invariant factors of M/N by applying the algorithm in Smith Normal Form to the matrix given by a set of generators for N. - Theorem about existence of basis for a free module M that is compatible with a submodule (Theorem 4 in Sec. 12.1 in DF) - How to prove Theorem 4 (in Sec. 12.1) "existence" part of F.T. using Smith Normal Form - Definition of annihilator of a submodule - Example of R=Z (F.T. of f.g. abelian groups) - Example of R=F[x] and how to prove existence of Jordan canonical form from F.T. (elementary divisor form). |
Course Information
o Time: MWF 11:00AM - 11:50AM o Location: Thackeray 525 o There will a midterm and a final exam. o There are weekly homework problems. I will ask you to hand in a few of the problems each week. I will randomly pick one or more assigned problems and grade them. o Midterm Friday Nov. 9 in class. |
Instructor's
Information
Name: Kiumars Kaveh Email: kavehk AT pitt.edu Office: Thackeray #424 Office phone: 412-624-8331 o Office hours: 3PM-4PM MW or by appointment. |
Homeworks and other online material
HW8: DF Sec. 10.5: Problems 2, 3, 6, 7, 9, 26 DF Sec. 12.1: Problems 1, 2, 3, 5, 6, 13 Problem: Let G be a finite group and and consider the group ring R=CG where C is the field of complex numbers. Show that any (left) R-module is both injective and projective. Hint: It is enough to show any submodule N of an R-module M has a complement, i.e. there is a submodule L of M such that direct sum of N and L is M. Please hand in the above problem and problems Sec. 10.5: 6, 9, 26, Sec. 12.1: 13 due Wednesday November 28 in class HW7: DF Sec. 10.3: Problems 2, 3, 5, 7-13, 18, 20, 22, 25, 27 DF Sec. 10.4: Problems 3, 5, 6, 9, 16, 11, 12, 19, 24-26 Please hand in Sec. 10.3: 2, 25, 27, Sec. 10.4: 5, 9, 11 due Monday November 5 in class HW6: Problem: Show that J is a monomial ideal if and only if the following condition holds: a polynomial f is in J if all its monomials are in J. DF Sec. 9.6: Problems 1, 2, 3, 4, 5, 9, 10, 11, 12, 14, 15 DF Sec. 10.1: Problems 5, 8, 9, 15 DF Sec. 10.2: Problems 3, 4, 6, 7, 9, 41 Please hand in the above problem and Sec. 9.6: 2, 5, 14 Sec. 10.1: 8, Sec. 10.2: 9 due Friday October 26 in class HW5: DF Sec. 9.2 : Problems 2, 3, 4, 7 DF Sec. 9.3 : Problems 1, 2, 4 DF Sec. 9.4 : Problems 1, 2, 3, 6, 7, 8, 10, 12, 15, 19 DF Sec. 9.5 : Problems 1, 3, 5 Please hand in Sec. 9.2: 4, Sec. 9.3: 4, Sec. 9.4: 3, 8, 15, Sec. 9.5: 3 due Wednesday October 17 in class HW4: DF Sec. 8.2 : Problems 1, 3, 4, 5, 8 DF Sec. 8.3 : Problems 2, 4, 5, 6, 7, 8 DF Sec. 9.1 : Problems 4, 6, 8, 14, 17, 18 Please hand in Sec. 8.2: 5, 8, Sec. 8.3: 8 and Sec. 9.1: 8, 17 due Friday October 5 in class HW3: Problem: Let R be an integral domain and S a multiplicative set in R. Show that there is a one-to-one correspondence between prime ideals in R that do not intersect S and prime ideals in S^{-1}R. Problem: Let R be an integral domain and S a multiplicative set in R. Let I be an ideal that does not intersect S. Show that I can be enlarged to a prime ideal P in R such that P does not intersect S either. (Hint: you can use the previous problem or you can prove it directly using Zorn's lemma.) DF Sec. 7.6 : Problems 3, 5, 7, 8, 9, 10, 11 DF Sec. 8.1 : Problems 3, 6, 8, 9, 10 Please hand in the above two problems and Sec. 7.6: 7, 11(d),(e) and Sec. 8.1: 3, 8(a) D=-2, 8(b) D=-43 due Friday September 28 in class HW2: DF Sec. 7.3 : Problems 2, 6, 7, 12, 13, 17, 18, 19, 24, 27, 34 DF Sec. 7.4 : Problems 2, 7, 8, 9, 11, 13, 15, 31, 32, 33, 37, 40 DF Sec. 7.5 : Problems 1, 3, 5 Please hand in Sec. 7.3: 13, 34 and Sec. 7.4: 15, 33 and Sec. 7.5: 5 due Friday September 21 in class HW1: DF Sec. 7.1 : Problems 14, 15, 21, 23, 25, 26 DF Sec. 7.2 : Problems 3, 5, 9, 13(a) Please hand in Sec. 7.1: 15, 23 and Sec. 7.2: 5, 13(a) due Wednesday September 12 in class |