Algebra II (Fall 2018)
University of Pittsburgh

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