Theorem A (modules) Let M and N be modules, and let φ : M → N be a module homomorphism. Then: The kernel of φ is a submodule of M, The image of φ is a submodule of N, and The image of φ is isomorphic to the quotient module M / ker(φ). In particular, if φ is surjective then N is isomorphic to M / ker(φ).
Theorem A (modules) Let M and N be modules, and let φ : M → N be a module homomorphism. Then: The kernel of φ is a submodule of M, The image of φ is a submodule of N, and The image of φ is isomorphic to the quotient module M / ker(φ). In particular, if φ is surjective then N is isomorphic to M / ker(φ).