MWF 11:3012:25 CL room 115, TR 11:3012:25 140 W 19th ave, room 136
Instructor: Sasha Leibman  office: MW406 office hours: Wednesday 23pm, Thursday 12pm 
email: leibman.1@math.osu.edu  
phone: 6142920663 
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 343345);
10.2.1, 2, 3, 7, 8 (page 350); 10.3.2, 3, 4, 5, 6, 7, 8, 9, 10 (pages 356358). 
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 356357).  
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 356358). 
written part:  due by Tuesday, January 23   Exercises 
10.2.6, 11 (page 350);
10.3.15, 22(ac) (pages 357358),
and the following problem:
Let M be an Rmodule and let I, J be ideals in R. (a) Prove that Ann(I+J)=Ann(I)∩Ann(J). (b) Prove that Ann(I∩J)⊇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(I∩J)=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 375377). 
written part:  due by Tuesday, January 30   Exercises  10.4.9, 10(a,b), 16(a,b), 20, 21(a,b) (pages 376377).  
HW4  recommended:  by Friday, February 2   Exercises 
10.5.19 (pages 403404);
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 422423);
11.3.1, 2, 4, 5 (page 435). 
written part:  due by Wednesday, February 14   Exercises  11.2.11(ac) (page 423); 11.3.3(af) (page 435).  
HW6  recommended:  by Friday, February 16   Exercises  11.5.1, 3, 4, 5, 7, 9, 13 (pages 454455). 
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 468472). 
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. 10.3.911)
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 10.3.1617)
Internal direct sums of submodules (Section 10.3) The pprimary 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 contravariant 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 12.1.1619)
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) 