Quotient by kernel is isomorphic to image

For a homomorphism f, the induced map M/ker(f) → im(f) is an isomorphism.

Quotient by kernel is isomorphic to image

Quotient by kernel is isomorphic to image: Let $f:M\to N$ be an $R$ -module homomorphism. Then the induced map

\bar f: M/\ker(f)\longrightarrow \operatorname{im}(f),\qquad \bar f(m+\ker(f))=f(m),

is a well-defined isomorphism of $R$ -modules.

This is the usual “image–kernel” form of the first isomorphism theorem for modules , relating quotient modules to kernels and images .