MATH:5010 - Spring 2017

Homework Assignment 1

Reading (from Dummit and Foote):

Read Sections 10.1, 10.2, 10.3.

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 24, at the beginning of the discussion section):

Section 10.1: # 5, 9, 10. (Note R is a ring with 1 and M is a unital left R-module. The term "R-module" means "unital left R-module".)
Section 10.2: # 4. (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: You need to show that the isomorphism in the third sentence of the problem is an isomorphism of Z-modules, i.e. an isomorphism of abelian groups.
Section 10.3: # 7.

Part B Problems (to be handed in on Friday, January 27, at the beginning of the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, January 24, 2 extra points will be added to your homework score.

Section 10.1: # 8, 19. (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".)
Hint for #19: Here you look at the F[x]-module (V,T) where V and T are as in the statement of the problem. As we discussed in class last semester, the F[x]-submodules of (V,T) are precisely those of the form (W,T_W) where W is an F-subspace of V with T(W) ⊆ W and T_W is the restriction of T to W.
Section 10.2: # 9, 12. (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: In #12, do NOT use #11 but prove this from scratch. The isomorphism is an isomorphism of R-modules.
Section 10.3: # 2.
Hint for #2: When n=m, there is nothing to show since then Rⁿ = R^m. In the other direction, follow the hint given. But note the following: If f: Rⁿ --> R^m is an R-module isomorphism, you need to show that f(I Rⁿ) =I R^m and deduce that f induces an R-module isomorphism Rⁿ/ I Rⁿ --> R^m/ I R^m.

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

Section 10.1: # 2, 4, 6, 7, 15, 18, 20, 21.
Section 10.2: # 1, 2, 3, 6, 7, 8, 10, 11.

Frauke Bleher
Jan 9 2017