Htykuut

I found that $operatorname{Ext}_{Bbb Z}^1(Bbb Z_9, Bbb Z) cong Bbb Z_9$ by calculating it as the quotient of two $operatorname{Hom}$-sets.

I am tasked with finding the (equivalence classes of) exact sequences corresponding to the elements in the group above.

I know two, the trivial one with middle module $Bbb Z oplus Z_9$, and the one with middle module $Bbb Z$ with morphisms $mu: Bbb Z to Bbb Z$, $mu(n) = 9n$ and $phi:Bbb Z to Bbb Z_9$, $phi(m) = bar{m}$.

Since $9$ is not prime I don't know what I need to construct the remaining seven exact sequences.

Do you know what a pushout is? The construction you need is

$$
begin{array}{llllllll}
0 & to & mathbb{Z} & stackrel{times 9}to & mathbb{Z} & to & mathbb{Z}/9 & to & 0 \
& & downarrow phi & & downarrow & & downarrow textrm{id} & & \
0 & to & mathbb{Z} & to & P & to & mathbb{Z}/9 & to & 0
end{array}
$$
where $P$ is the pushout of $times 9$ and $phi$, the homomorphism whose ext-class you are working with.

For abelian groups, the pushout can be identified with the quotient of the direct sum $mathbb{Z}oplus mathbb{Z}$ by the subgroup generated by things of the form $(phi(z), -9z)$: you can see immediately that $phi=0$ gives you the trivial extension you mentioned.