Editing Isomorphism theorems (section)

== Modules ==
The statements of the isomorphism theorems for [[module (mathematics)|modules]] are particularly simple, since it is possible to form a [[quotient module]] from any [[submodule]].  The isomorphism theorems for [[vector space]]s (modules over a [[field (mathematics)|field]]) and [[abelian group]]s (modules over <math>\mathbb{Z}</math>) are special cases of these.  For [[dimension (vector space)|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 <math>M</math> and <math>N</math> be modules, and let <math>\varphi:M\rightarrow N</math> be a [[module homomorphism]].  Then:
# The [[kernel (algebra)|kernel]] of <math>\varphi</math> is a submodule of <math>M</math>,
# The [[image (mathematics)|image]] of <math>\varphi</math> is a submodule of <math>N</math>, and
# The image of <math>\varphi</math> is [[Module_homomorphism#Terminology|isomorphic]] to the [[quotient module]] <math>M/\ker\varphi</math>.
In particular, if <math>\varphi</math> is surjective then <math>N</math> is isomorphic to <math>M/\ker\varphi</math>.

===Theorem B (modules)===
Let <math>M</math> be a module, and let <math>S</math> and <math>T</math> be submodules of <math>M</math>.  Then:
# The sum <math>S+T=\{s+t\mid s\in S,t\in T\}</math> is a submodule of <math>M</math>,
# The intersection <math>S\cap T</math> is a submodule of <math>M</math>, and
# The quotient modules <math>(S+T)/T</math> and <math>S/(S\cap T)</math> are isomorphic.

===Theorem C (modules) ===
Let ''M'' be a module, ''T'' a submodule of ''M''.

# If <math>S</math> is a submodule of <math>M</math> such that <math>T \subseteq S \subseteq M</math>, then <math>S/T</math> is a submodule of <math>M/T</math>.
# Every submodule of <math>M/T</math> is of the form <math>S/T</math> for some submodule <math>S</math> of <math>M</math> such that <math>T \subseteq S \subseteq M</math>.
# If <math>S</math> is a submodule of <math>M</math> such that <math>T \subseteq S \subseteq M</math>, then the quotient module <math>(M/T)/(S/T)</math> is isomorphic to <math>M/S</math>.
<!-- We also need to mention the isomorphism theorems for topological vector spaces, Banach algebras etc. -->

===Theorem D (modules)===
Let <math>M</math> be a module, <math>N</math> a submodule of <math>M</math>. There is a bijection between the submodules of <math>M</math> that contain <math>N</math> and the submodules of <math>M/N</math>. The correspondence is given by <math>A\leftrightarrow A/N</math> for all <math>A\supseteq N</math>. This correspondence commutes with the processes of taking sums and intersections (i.e., is a [[lattice isomorphism]] between the lattice of submodules of <math>M/N</math> and the lattice of submodules of <math>M</math> that contain <math>N</math>).<ref>Dummit and Foote (2004), p. 349</ref>