## Description

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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