Math 5591H and 5112, Algebra II

MWF 11:30-12:25 CL room 115, TR 11:30-12:25 140 W 19th ave, room 136

Instructor: Sasha Leibman office: MW406     office hours: Wednesday 2-3pm, Thursday 1-2pm
e-mail: leibman.1@math.osu.edu
phone: 614-292-0663

Syllabus

Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition

Midterm 1 on Wednesday, Fabruary 7. Solutions. List of topics. Sample problems.

Homework:

HW1 recommended: by Friday, January 12 -- Exercises 10.1.1, 2, 3, 5, 6, 7, 11, 12, 13, 18, 19, 20, 22 (pages 343-345);
10.2.1, 2, 3, 7, 8 (page 350);
10.3.2, 3, 4, 5, 6, 7, 8, 9, 10 (pages 356-358).
written part: due by Tuesday, January 16 -- Exercises 10.1.8(a,b,c), 9, 10 (page 344); 10.2.9 (page 350); 10.3.7, 12(a,b) (pages 356-357).

HW2 recommended: by Friday, January 19 -- Exercises 10.2.4, 5, 10, 12 (page 350);
10.3.13, 14, 20, 21, 23, 24, 27 (pages 356-358).
written part: due by Tuesday, January 23 -- Exercises 10.2.6, 11 (page 350); 10.3.15, 22(a-c) (pages 357-358), and the following problem:
Let M be an R-module and let I, J be ideals in R.
(a) Prove that Ann(I+J)=Ann(I)∩Ann(J).
(b) Prove that Ann(IJ)⊇Ann(I)+Ann(J).
(c) Give an example where the inclusion in (b) is strict.
(d) If R is commutative and unital and I, J are comaximal, prove that Ann(IJ)=Ann(I)+Ann(J).

HW3 recommended: by Friday, January 26 -- Exercises 10.4.2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 18, 19, 24, 25 (pages 375-377).
written part: due by Tuesday, January 30 -- Exercises 10.4.9, 10(a,b), 16(a,b), 20, 21(a,b) (pages 376-377).

HW4 recommended: by Friday, February 2 -- Exercises 10.5.1-9 (pages 403-404);
11.5.4, 6, 8 (page 455).
No written part.

HW5 recommended: by Monday, February 12 -- Exercises 11.2.1, 2, 3, 5, 6, 7, 9, 10 (pages 422-423);
11.3.1, 2, 4, 5 (page 435).
written part: due by Wednesday, February 14 -- Exercises 11.2.11(a-c) (page 423); 11.3.3(a-f) (page 435).

HW6 recommended: by Friday, February 16 -- Exercises 11.5.1, 3, 4, 5, 7, 9, 13 (pages 454-455).
written part: due by Tuesday, February 20 -- Exercises 11.5.5, 12(a,b,c) (page 455); 11.4.2, 3 (page 441).

HW7 recommended: by Friday, February 23 -- Exercises 12.1.1, 2, 3, 5, 6, 8, 10, 13, 14, 20, 21, 22 (pages 468-472).
written part: due by Tuesday, February 27 -- Exercises 12.1.4, 9, 11, 12(a,b) (page 469).

My solutions to HW1, HW2, HW3, HW5, HW6

Studied topics:
January 8: Definition and examples of modules (Section 10.1)
January 9: Homomorphisms of modules (Section 10.2)
January 10: Simple modules. Schur's lemma (ex. 10.3.9-11)
Factorization of modules (Section 10.2)
Isomorphism theorems for modules (Section 10.2)
Generating sets of modules (Section 10.3)
January 11: The universal property of free modules (Section 10.3)
Direct products and sums of modules. The universal property of direct products and sums (Section 10.3)
January 12: Exercises from Section 10.1.
January 16: Infinite direct products and sums and their universal prperties.
January 17: The Chinese remainder theorem for modules (execises 10.3.16-17)
Internal direct sums of submodules (Section 10.3)
The p-primary components of a module (execises 10.3.18,22)
Free modules (Section 10.3)
January 18: "Internal" free modules and bases (Section 10.3)
Vector spaces are free modules and their rank is well defined (Section 11.1)
January 19: Exercises from Section 10.3.
January 22: Maximal linear independent subsets in general modules (exercise 12.1.2)

Definition of the tensor product of modules over a commutative ring (Section 10.4)
January 23: Properties of tensor product (Section 10.4)
Tensor product of direct sums (Section 10.4)
January 24: Tensor product of a module and of a quotient of the ring (Section 10.4)
Tensor product of a module and of the field of fractions (Section 10.4)
January 25: Extension of scalars (Section 10.4)
Tensor product of algebras (Section 10.4)
Tensor product of homomorphisms (Section 10.4)
January 26: Exercises from Section 10.4.
January 29: The tensor algebra of a module. The symmetric and the exterior algebras (Section 11.5)
January 30: Commutative diagrams and exact sequences of modules (Section 10.5)
The short five lemma (Section 10.5)
January 31: Flat modules (Section 10.5)
February 1: Projective modules (Section 10.5)
February 2: Injective modules (Section 10.5)
February 5: Review.
February 6: Review.
February 7: Midterm 1.

February 8: Homomorphisms of free modules of finite rank. Matrices of homomorphisms (Section 11.2)
Change of basis and transition matrices (Section 11.2)
February 9: Change of matrices under a basis change. Similar matrices. (Section 11.2)
Dual modules, dual bases, and dual homomorphisms (Section 11.3)
February 12: Finite dimensional vector spaces (Section 11.2)
Exercises from Section 11.2.
February 13: Exercises from Sections 11.2, 11.3.
February 14: Contraction of tensors. The trace.
Homomorphisms as tensors. Matrix multiplication.
Co- and contra-variant tensors.
February 15: The symmetric tensor algebra and the exterior algebra of a free module (Section 11.5)
February 16: Symmetric and alternating tensors (Section 11.5)
The determinant of a linear transformation (Section 11.4)
February 19: Symmetric and alternating tensors (Section 11.5)
February 20: Properties of the determinant of a linear transformation (Section 11.4)
"Canonical forms" of a matrix of a linear transformation.
The rank of a module over an integral domain (Section 12.1)
February 21: Submodules of free modules of finite rank over a PID (Section 12.1)
February 22: Finding bases of submodules of free modules of finite rank over EDs (Exercises 12.1.16-19)
The fundamental theorem of modules of finite rank over PIDs. Invariant factors. (Section 12.1)
February 23: Exercises from Section 12.1.
Elementary divisors of modules of finite rank over PIDs (Section 12.1)

Plans for the nearest future:
Uniqueness of invariant factors (Section 12.1)
Finding the invariant factors; the relations matrix (Section 12.1)
The rational canonical form of the matrix of a linear transformation of a finite dimensional vector space (Section 12.2)
The Smith normal form of a matrix (Section 12.2)
The minimal and the characteristic polynomials of a transformation (Section 12.2)
The Jordan normal form of a matrix (Section 12.3)