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Math 512 Problem Set 4 

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Math 512 Problem Set 4
Exercise 1. If A is a finite abelian group, show that A ⊗Z Q = 0.
Exercise 2. Show that Zm ⊗Z Zn
∼= Zd, where d = (m, n). Hint: Write
d = am + bn for some integers a, b.
Exercise 3. Let M be an R-module, and I ⊂ R an ideal. Show that
R/I) ⊗R M ∼= M/IM.
Exercise 4. Let R be commutative, and I, J ⊂ R ideals. Show that
R/I ⊗R R/J ∼= R/(I + J).
Exercise 5. Let R be commutative. An R-module is called flat if tensoring
with that module is left exact. Show that every projective R-module is flat.
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