First isomorphism theorem for modules

A module homomorphism induces an isomorphism M/ker f ≅ im f.

First isomorphism theorem for modules

First isomorphism theorem (modules): Let $f\colon M\to N$ be a module homomorphism . Then the induced map

\overline f\colon M/\ker(f)\longrightarrow \mathrm{im}(f),\qquad \overline f(m+\ker(f))=f(m),

is an isomorphism of $R$ -modules. Here $\ker(f)$ is the kernel and $\mathrm{im}(f)$ is the image , and $M/\ker(f)$ is a quotient module .

This theorem identifies the “effective domain” of a map with its image and is the basic mechanism behind many results in the theory of exact sequences ; it is the module analogue of the first isomorphism theorem for rings

Proof sketch: Define $\overline f$ as above; it is well-defined because elements differing by $\ker(f)$ have the same image. Surjectivity is immediate from the definition of $\mathrm{im}(f)$ , and injectivity follows because $\overline f(m+\ker(f))=0$ implies $m\in\ker(f)$ . Linearity is inherited from $f$ .