1. ## Short exact sequences

I am claiming the following is not a short exact sequence
$\displaystyle 0\rightarrow Z\stackrel{i}{\rightarrow} Z \oplus Z \oplus Z \stackrel{j}{\rightarrow} Z \rightarrow 0$

I am thinking of it this way: If it was a short exact sequence, then i would be injective and j would be onto. Also, $\displaystyle (Z \oplus Z \oplus Z )/ i(Z)$ would be isomorphic to $\displaystyle Z$ .

Now $\displaystyle i(1)= (a,b,c) \neq 0$ for some $\displaystyle a,b,c \in Z$. Now, is it true that
$\displaystyle i(Z)=im(i)=Z<(a,b,c)>$ ? And that $\displaystyle (Z \oplus Z \oplus Z )/ i(Z) = Z/aZ \times Z/bz \times Z/cZ$?If yes, how do you show this is not isomorphic to $\displaystyle Z$ ? If not, what am I doing wrong and how do you prove my claim?

2. That sequence can't be exact. If you have 0->A->B->C->0, then C is isomorphic to B/A', where A' is the image of A under the map A->B. Because of exactness, that map is injective, so that A' is isomorphic to A. You can't take Z^3, mod out by something isomorphic to Z, and get Z.

3. Ok, that makes sense, but what would be a good argument to show $\displaystyle (Z \oplus Z \oplus Z) / A'$ is not isomorphic to $\displaystyle Z$?

4. Can we say, $\displaystyle } Z \oplus Z \oplus Z$ has 3 generators, now $\displaystyle ( Z \oplus Z \oplus Z)/i(Z)$ is isomorphic to $\displaystyle ( Z \oplus Z \oplus Z) / Z$ which has at least 2 generators, but $\displaystyle Z$ only has one generator?

5. You have to be careful when talking about the number of generators, because it's not fixed--for instance, the integers are generated by (1) or by (2,3). I think the free rank of an abelian group might be helpful here.

6. Or just say it's not exact by definition. To be exact, we would need $\displaystyle \mathrm{im}(i)=\mathrm{ker}(j)$.

Of course, the other issue is still a good question in and of itself, so don't let this stop that discussion.

7. I am not sure I understand how $\displaystyle im(i) = Ker(j)$ helps show $\displaystyle Z^3/i(Z)$ is not isomorphic to $\displaystyle Z$. Can you elaborate on that?

Also @ Tinyboss: How, do I use the free rank of an abelian group to prove that? My algebra is very rusty, and I tried to look it up, but I don't think I understand how to apply the definition here...

8. Sorry, what I thought would work, doesn't. I asked a buddy and he said to tensor the sequence with Q, yielding the same sequence except with Q's in place of Z's. Since Q is a field, there is a well-defined notion of dimension, and so that sequence can't be exact. And because tensoring preserves exact sequences, this implies the original sequence wasn't.

I don't understand it 100%, but I'm putting it here since my first guess was wrong.

9. Originally Posted by math8 I am not sure I understand how $\displaystyle im(i) = Ker(j)$ helps show $\displaystyle Z^3/i(Z)$ is not isomorphic to $\displaystyle Z$. Can you elaborate on that?

Also @ Tinyboss: How, do I use the free rank of an abelian group to prove that? My algebra is very rusty, and I tried to look it up, but I don't think I understand how to apply the definition here...
I don't know that it does. I was just giving another way of answering the original question, about the sequence not being exact.

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