**Problem 1**. Let be an abelian group. Prove that if has no element of infinite order, then

**Solution**. If and then for all we have

**Problem 2**. Let be an abelian group. Prove that if has an element of infinite order, then

**Solution**. Let be an element of infinite order in Since is a flat -module, is a subgroup of So we only need to prove that This is obvious because, since has infinite order, we have and thus

Let be a commutative ring with unity and let be a set which may or may not be finite. Suppose that are -modules. Let be an -module. Recall that tensor product distributes over direct sum, i.e.

as -modules. Next problem shows that the above is not true in general if is replaced with

**Problem 3**. Let Prove that

**Solution**. Each is a finite abelian group and thus by Problem 1. Thus

Now let Let Suppose that the order of is finite, say Then for all That means divides for all which is nonsense. So the order of in is infinite and hence, by Problem 2,

The result now follows from (1) and (2).