Category Kernel Question (Unique homomorphism $\alpha_0:\ker\phi\to L$)

Question

There is a a question with two parts (prove or disprove), one of which I know how to do, but the other part I am stuck.

Let $\phi:M'\to M$ be a homomorphism of abelian groups. Suppose that $\alpha:L\to M'$ is a homomorphism of abelian groups such that $\phi\circ\alpha=0$, for instance the inclusion map $\mu:\ker\phi\to M$. Prove or disprove each of the following:

1. There is a unique homomorphism $\alpha_0:\ker \phi\to L$ such that $\mu=\alpha\circ\alpha_0$.

   (This I am unsure if it is true or untrue)

2. There is a unique homomorphism $\alpha_1:L\to\ker\phi$ such that $\alpha=\mu\circ\alpha_1$.

   (This I am quite sure it is true as it is the universal property of the kernel.)

I will post my solution to part 2 below.

Answer 1

Part (i) is false: Take $M=L=0$ and $M'=\Bbb{Z}/2\Bbb{Z}$ so that there are unique homomorphisms $$\phi:\ M'\ \longrightarrow\ M\qquad\text{ and }\qquad\alpha:\ L\ \longrightarrow\ M',$$ and they clearly satisfy $\phi\circ\alpha=0$. Then $\ker\phi=M'$ so the inclusion map $\mu:\ \ker\phi\ \longrightarrow\ M'$ is the identity on $M'=\Bbb{Z}/2\Bbb{Z}$, which cannot factor over $L=0$, i.e. we cannot have $\mu=\alpha\circ\alpha_0$.

Part (ii) is precisely the usual categorical definition of the kernel.

Answer 2

Part (ii) is true.

Uniqueness: Suppose $\alpha=\mu\circ\alpha_1=\mu\circ\beta$. Then $\mu\alpha_1(x)=\mu\beta(x)$.

This means that $\mu(\alpha_1(x)-\beta(x))=0$. Hence $\alpha_1(x)-\beta(x)=0$ and $\alpha_1(x)=\beta(x)$.

Existence: Take $\alpha_1:L\to \ker\phi$, $\alpha_1(l)=\alpha(l)$.

Since $\phi(\alpha(l))=0$, thus $\alpha(l)\in\ker\phi$.

$\alpha_1(l_1+l_2)=\alpha(l_1+l_2)=\alpha(l_1)+\alpha(l_2)=\alpha_1(l_1)+\alpha_1(l_2)$ hence it is a homomorphism.

$\mu\alpha_1(l)=\mu\alpha(l)=\alpha(l)$.