A section of the quotient map or a retraction of the inclusion splits a short exact sequence. The point is to make the formal expression readable enough to audit line by line.
The mathematical object
An \(R\)-module generalises both vector spaces and abelian groups. A map \(f:M\to N\) is understood through its kernel, image, and cokernel, which exact sequences place on one line.
I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).
One explicit computation
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
Why the identity matters
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
Where it can fail
Tensor product is right exact but not generally left exact, while \(\operatorname{Hom}_R(P,-)\) is exact precisely when \(P\) is projective.
This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.