MATH:5010 - Spring 2017

Homework Assignment 2

Reading (from Dummit and Foote):

Read Sections 10.3, 10.4 (pp. 364-366, examples (1)-(5) on pp. 368-369), 10.5 (pp. 378-385).

Please look through the sections for EACH lecture before the lecture. See the Lecture Schedule for the sections that will be covered.

Then after EACH lecture, review your lecture notes and thoroughly read the sections covered in class.

Part A Problems (to be handed in on Tuesday, January 31, at the beginning of the discussion section):

Section 10.3: # 5, 15. (Note R is a ring with 1 and M is a unital left R-module. The term "R-module" means "unital left R-module".)
IMPORTANT for #5: A torsion R-module is an R-module M with M = Tor(M) (see Section 10.1 #8). For the example, take as R the integers Z and let T be the (infinite) direct sum of Z/nZ over all positive integers n. Prove that T is a torsion module whose annihilator is the zero ideal. (Of course you can come up with a different example, but it should be a torsion module over an integral domain.)
Hint for #15: The direct sum is an internal direct sum, so you need to verify that eM and (1-e)M are submodules of M, M = eM + (1-e)M and the intersection of eM and (1-e)M is {0}.
Section 10.4: # 16. (Note R is assumed to have a 1, and "R-module" means again "unital R-module".)
GENERAL REMARK: When you need to show that a module homomorphism is bijective for which the domain is a tensor product, it is often not so easy to prove the map is injective (because, in general, tensors can be written in many different ways as finite sums of simple tensors). Hence the usual approach is to find an inverse map. For #16, you can prove injectivity in part (b) directly because of part (a), but you can also go for an inverse map.
Section 10.5: # 2.
IMPORTANT: This is a commutative diagram of GROUPS. So please write everything MULTIPLICATIVELY. In particular, the identities in the groups should be written as 1 and not as 0. The proof is very similar to our proof of the Short Five Lemma in class, but you need to write your arguments multiplicatively.

Part B Problems (to be handed in on Friday, February 3, at the beginning of the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, January 31, 2 extra points will be added to your homework score.

Section 10.3: # 9, 11, 12 (see below). (Note R is a ring with 1≠0 and M is a unital left R-module. The term "R-module" means "unital left R-module".)
IMPORTANT for #9, #11: An irreducible R-module is also called a simple R-module.
Hint for #9: To find all irreducible Z-modules, use that Z-modules are additively written abelian groups. The definition of irreducible Z-modules shows that they are the same as (additive) abelian groups which have no proper nonzero subgroups. The first part of the problem also shows that they are all cyclic.
IMPORTANT for #12: Hand in either (a) or (b). You should be able to do both parts, but you only need to hand in one of them.
Section 10.4: # 20. In fact I want you to hand in solutions to the following steps:
Prove that there is an R-balanced map f: I x I --> I² defined by f(p(x),q(x)) = p(x)q(x). Use this to obtain a well-defined Z-module homomorphism F : I ⊗_R I --> I². Argue that F is surjective. Show that F is an R-module homomorphism by showing F preserves the R-module structure on I ⊗_R I coming from the fact that R is commutative. Then use the map F to do #20.

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

Section 10.3: # 1, 4, 6, 10, 13, 14, 16, 17, 20, 23.
(Look at #27 to see an example where the rank of a free R-module is not well defined when R is not commutative.)
Section 10.4: # 2, 3, 4, 6, 7, 11, 17.
Section 10.5: # 1, 27.

Frauke Bleher
Jan 24 2017