I am trying to solve the following question:
Let $$ 0 \to_0R \to_i M \to_j N \to_0 0 $$ be a sequence of rings and homomorphisms such that $i$ is injective, $j$ is surjective and $im(i)=ker(j)$. Prove that the following are equivalent:
- $M \cong R \oplus N$ with $i : R \to R \oplus N$ and $ j : R \oplus N \to N$ the canonical homomorphisms;
- There is a homomorphism $q : M \to R$ such that $q \circ i = 1$;
- There is a homomorphism $ p : N \to M $ such that $j \circ p = 1 $;
I actually have couple of questions in my attempt:
My Attempt:
I am trying to prove the following implications : (1) $\implies$ (2), (2) $\implies$ (1), (1) $\implies$ (3) and (3) $\implies$ (1).
I am done with (1) $\implies$ (2) & (3). But while going backword, I stuck in the surjectivity part. For example, solving $ (2) \implies (1) $, I defined the map $$\phi : M \to R \oplus N$$ $$ \phi(m) = ( q(m),j(m) )$$ This map is well defined and homomorphism. I have also proved that it is injective, but while showing it is surjective, I stuck. For example, TO prove it is surjective, I need to show that for any $(r,n)$, there exist a $m \in M$ such that $\phi(m)=(r,n)$ i.e $ q(m)=r $ and $ j(m) = n $.
Now, since $j$ is surjective, so for every $n$, there exist a $m \in M$ such that $j(m)=n$, Now the problem is, I am unable to show that for the same $m$, $q(m)=r$ ??
In the implication $(3) \implies (1)$, I defined the following map (that looks the natural one) $$ \psi : R \oplus N \to M $$ $$ (r,n) \mapsto i(r) + p(n) $$
This map is weldefined. It preserves Addition but there is a problem in multiplication. Am I defining the map in the right way?? This map is injective but again I am confused at surjectivity part.
My Questions?
- How to prove the surjectivity of map $\phi$?
- In $(3) \implies (1)$, is the map I defined would work for proving isomorphism? If so, then how to show it preserves multiplication?
- The surjectivity of map $\psi$ ?
any kind of help would be really appreciable. Thanks in Advance.
