If $M,N$ are $R$-modules and $f:Μ\rightarrow N$ , $g:N\rightarrow M$ are $R$-module homomorphisms s.t. $g\circ f=1_M$ Prove: $N=Im(f)\oplus \ker(g)$

Question

Question: If $M,N$ are $R$-modules and $f:Μ\rightarrow N$ , $g:N\rightarrow M$ are $R$-module homomorphisms s.t. $g\circ f=1_M$

Prove: $N=Im(f)\oplus \ker(g)$

My attempt: Since $\forall n\in Im(f)$ we get $g(n)=1_M\neq 0_M$ $\implies Im(f)\cap \ker(g)=\emptyset$

In addition $Im(f)\subseteq N$ and $\ker(g)\subseteq N \implies Im(f)\oplus \ker(g)\subseteq Ν $

However i am having trouble proving that $N\subseteq Im(f)\oplus \ker(g)$

How should i proceed?

Note that $1_M$ here denotes the identity function on $M$, defined as $1_M(m)=m$ and we don't have such an element as $1_M$ in a general module. — Berci, Aug 23 '20 at 14:07

Surb · Accepted Answer · 2020-08-23T14:19:49.037

1

Hint

I guess that $1_M$ means the identity function on $M$. If $n\in N$, then $$n=\underbrace{n-f\big(g(n)\big)}_{\in \ker(g)}+\underbrace{f\big(g(n)\big)}_{\in \text{Im(f)}}.$$

edited Aug 23 '20 at 14:19

answered Aug 23 '20 at 13:32

Surb

55,662

If $M,N$ are $R$-modules and $f:Μ\rightarrow N$ , $g:N\rightarrow M$ are $R$-module homomorphisms s.t. $g\circ f=1_M$ Prove: $N=Im(f)\oplus \ker(g)$

1 Answers1