Math 512 Problem Set 2
Exercise 1. Let R be a commutative unital ring. Show that
Mn = {p ∈ R[x] | deg p < n}
is a submodule of R[x].
Exercise 2. Let M be an R-module, and I ⊂ R an ideal.
1. Show that IM = {
Pn
i=1 rimi
| n ∈ N, ri ∈ I, mi ∈ M} is a submodule
of M.
2. Show that M/IM is an R/I module, with multiplication given by
(r + I)(m + IM) = rm + IM for all r + I ∈ R/I, m + IM ∈ M/IM.
Exercise 3. Prove the Five Lemma: Let
A1 −−−→ A2 −−−→ A3 −−−→ A4 −−−→ A5
yα1
yα2
yα3
yα4
yα5
B1 −−−→ B2 −−−→ B3 −−−→ B4 −−−→ B5
be a commutative diagram of R-module homomorphisms with each row exact.
(a) Show that if α1 is surjective, and α2 and α4 are injective, then α3 is also
injective.
(b) Show that if α5 is injective, and α2 and α4 are surjective, then α3 is also
surjective.
Exercise 4. Let f : A → A be an R-module homomorphism. Show that if
ff = f, then A ∼= ker f ⊕ Im f.
Exercise 5. Let f : A → B and g : B → A be R-module homomorphisms.
Show that if gf = id, then B ∼= Im f ⊕ ker g.
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