, then is a subring of . Every subring of is of the form for some subring of such that . If is an ideal of such that , then is an ideal of . Every ideal of is of the form for some ideal of such that . If is an ideal of such that , then the quotient ring is isomorphic to . Theorem D (rings) Let be an ideal of . The correspondence is an inclusion-preserving bijection between the set of subrings of that contain and the set of subrings of . Furthermore, (a subring containing ) is an ideal of if and only if is an ideal of . Modules The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over ) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem. In the following, "module" will mean "R-module" for some fixed ring R. 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 B (modules) Let M be a module, and let S and T be submodules of M. Then: The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M, The intersection S ∩ T is a submodule of M, and The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic. Theorem C (modules) Let M be a module, T a submodule of M. If is a submodule of such that , then is a submodule of . Every submodule of is of the form for some submodule of such that . If is a submodule of such that , then the quotient module is isomorphic to . Theorem D (modules) Let be a module, a submodule of . There is a bijection between the submodules of that contain and the submodules of . The