1. There will be 6 problems: 2 easy, 2 intermediate, and 2 more challenging.

2. The topics will cover the material on rings, modules and vector spaces in the book discussed in the lecture/discussions, with focus on Sections 9.1-9.5, 10.1-10.3 and 11.1-11.3.

3. There will be no question asking to reproduce a proof of a theorem from the book, instead, the problems will ask to use the theory we went through so far to
(a) solve some given problems about concrete polynomial rings, modules and vector spaces.
(b) prove some simple general facts about polynomial rings, modules and vector spaces.
The problems will be similar to homework problems and to recommended problems suggested on the homework assignment website.

4. Here are some topics to focus on:
(a) Basic knowledge about concrete polynomial rings we went through, and ideals in these rings.
(b) Polynomial rings which are unique factorization domains.
(c) Irreducibility criteria.
(d) Basic facts about modules, their submodules, module homomorphisms and quotient modules.
(e) Generation of modules and free modules.
(f) Basic facts about vector spaces, their subspaces, vector spaces homomorphisms and quotient spaces.
(g) Linear independence, bases and the Exchange principle.
(h) Basic facts about matrices of vector space homomorphisms. Change of bases and the associated matrix.
(i) Dual spaces, dual bases, dual homomorphisms, the connection between the matrices of a vector space homomorphism and its dual, rank of a matrix.

5. It will be important that you express yourself clearly and correctly. It will be expected that you write your arguments rigorously, but include just the right amount of information. Please take this into account when preparing for the midterm.

M230