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

Related Questions

Action of the $n$-th roots of unity on $mathbb{A}^2$

1  Asked on November 19, 2021

MIN-FORMULA $in$ NP

1  Asked on November 19, 2021 by ettore

$forall xinmathbb{R}enspace enspace, forall yinmathbb{R} : (xge y)$ or $(xle y)$ proof?

1  Asked on November 19, 2021

Interchange of diagonal elements with unitary transformation

2  Asked on November 19, 2021

How can we prove mgf of sample proportion of binomial distribution converges to exp(pt)?

1  Asked on November 19, 2021

What is the difference between a measure being “supported on [set] $A$” vs “supported in $A$” vs “supported at $A$”?

0  Asked on November 18, 2021

Is taking the derivative inside this integral acceptable?

1  Asked on November 18, 2021

Moduli Space of Tori

1  Asked on November 18, 2021 by bookworm

Necessary condition for x>0 being an integer

2  Asked on November 18, 2021 by aeternal

Does the Null spaces of matrix $ntimes n$ matrix $A$ and matrix $BA$ equal to each other if the matrix $B$ is invertible?

1  Asked on November 18, 2021

Given a basis $mathcal{B}$, can I assume that $mathcal{B}$ is orthonormal?

3  Asked on November 18, 2021

Hausdorff and non-discrete topology on $mathbb{Z}$

3  Asked on November 18, 2021 by uday-patel

Rearrangement diverges then original series also diverges?

2  Asked on November 18, 2021 by akash-patalwanshi

Absolutely continuous function with bounded derivative on an open interval is Lipschitz

2  Asked on November 18, 2021

Under what condition on $A$ is the following true: $lambda_{min}(A) |x|_2^2 leq x^T Ax leq lambda_{max}(A) |x|_2^2$?

1  Asked on November 18, 2021

Convergence in L2 (up to a constant) implies convergence in probability?

1  Asked on November 18, 2021 by user3294195

Find the constant for $int_{0}^{1} {frac{mathrm{d}x}{sqrt{(1-x^2)(1-(kx)^4)}}} sim Cln(1-k)$

1  Asked on November 18, 2021 by nanayajitzuki

Prove $lim_{nmapsto 0}[(psi(n)+gamma)psi^{(1)}(n)-frac12psi^{(2)}(n)]=2zeta(3)$

2  Asked on November 18, 2021

Prove or disprove that If $amid c$ and $bmid c$, then $ab mid c$.

1  Asked on November 18, 2021 by mathyviking

Question on Vectors and Linear equation

1  Asked on November 18, 2021