MHB Each exact sequence can be arised by short exact sequences

[mathmari]

Hey!

Let $R$ be a commutative ring with unit.

We have that if the sequences $0\rightarrow A\rightarrow B\overset{f}{\rightarrow}C\rightarrow 0$ and $0\rightarrow C\overset{g}{\rightarrow}D\rightarrow E\rightarrow 0$ are exact, then the sequence $0\rightarrow B\overset{gf}{\rightarrow} D\rightarrow E\rightarrow 0$ is exact.

So, each exact sequence can be arised by short exact sequences as above, right? (Wondering)

But how could we prove this? Could you give me a hint? (Wondering)

 

[mathmari]

Do we show that maybe using induction on the length of a sequence?

Base case: We consider the sequence $0\rightarrow A\rightarrow B\rightarrow 0$. This is exact.

Inductive hypothesis: We assume that each exact sequnez of length $n$ is made by short exact sequences.

Inductive step: We consider a sequence of length $n+1$. The first $n$ are made by short exact sequences, because of the inductive hypothesis, right? How could we continue? (Wondering)

 

[steenis]

See exercise 2.6 on p.65 of Rotman - An Introduction to Homological Algebra 2nd edition 2009.