# Need help to understand a theorem about direct sums and regular morphism of R-modules

Mathematics Asked by Luiz Guilherme De Carvalho Lop on December 12, 2020

i dont know how show that theorem is true

Definition: A morphism $$F: M longrightarrow N$$ of $$R$$-modules is called regular if exist $$G: N longrightarrow M$$ such that $$F circ G circ F = F$$.

Theorem: $$F$$ a morphism of $$R$$-modules $$M$$ and $$N$$, is regular if and only if $$ker(F)$$ is a direct summand of $$M$$ and $$im(F)$$ is a direct summand of $$N$$.

the only "tip" is: choice good exact sequences and see when it split.

I shall work in $$R-Mod$$, the category of left $$R$$-modules. Suppose $$exists G:Nrightarrow M$$ such that $$Fcirc Gcirc F=F$$.

Consider the short exact sequence$$0 rightarrow operatorname{Ker}Fhookrightarrow Mxrightarrow{F} operatorname {Im}Frightarrow 0$$ Look at $$G|_{operatorname{Im}F}:operatorname{Im}Frightarrow M$$. We have $$Fcirc G|_{operatorname{Im}F}=id_{operatorname{Im}F}$$ by the condition. As such the above sequence is a split short exact sequence (it right splits and hence left splits as well). $$therefore operatorname{Ker } F$$ is a direct summand of $$M$$.

Consider the short exact sequence $$0rightarrow operatorname{Im}Fxrightarrow{j} N rightarrow operatorname{CoKer}Frightarrow 0$$
Look a $$Fcirc G:Nrightarrow operatorname{Im}F$$. Once again, we have $$Gcirc Fcirc j=id_{operatorname{Im}F}$$. So the above sequence left splits and hence is a split short exact sequence. $$therefore operatorname{Im}F$$ is a direct summand of $$N$$.

Now assume $$operatorname{Ker}F$$ is a direct summand of $$M$$. Then $$0 rightarrow operatorname{Ker}Fhookrightarrow Mxrightarrow{F} operatorname {Im}Frightarrow 0$$ is a split short exact sequence. So we get $$g:operatorname {Im}Frightarrow M$$ such that $$Fcirc g=id_{operatorname{Im}F}$$. We also have a sub-module $$N'$$ of $$N$$ such that $$N=operatorname{Im}Foplus N'$$ Define $$G:Nrightarrow M$$ $$(x,y)mapsto g(x)$$

Then $$Fcirc Gcirc F(x)=Fcirc G((F(x),0))=Fcirc g(F(x))=F(x)$$

Answered by Soumik on December 12, 2020

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