Editing Module (mathematics) (section)

== Submodules and homomorphisms ==

Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]] of ''M''.  Then ''N'' is a '''submodule''' (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product {{nowrap|''r'' ⋅ ''n''}} (or {{nowrap|''n'' ⋅ ''r''}} for a right ''R''-module) is in ''N''.

If ''X'' is any [[subset]] of an ''R''-module ''M'', then the submodule spanned by ''X'' is defined to be <math display="inline">\langle X \rangle = \,\bigcap_{N\supseteq X} N</math> where ''N'' runs over the submodules of ''M'' that contain ''X'', or explicitly <math display="inline">\left\{\sum_{i=1}^k r_ix_i \mid r_i \in R, x_i \in X\right\}</math>, which is important in the definition of [[tensor product of modules|tensor products of modules]].<ref>{{Cite web|url=http://people.maths.ox.ac.uk/mcgerty/Algebra%20II.pdf|title=ALGEBRA II: RINGS AND MODULES|last=Mcgerty|first=Kevin|date=2016}}</ref> 

The set of submodules of a given module ''M'', together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a [[Lattice (order)|lattice]] that satisfies the '''[[modular lattice|modular law]]''':
Given submodules ''U'', ''N''<sub>1</sub>, ''N''<sub>2</sub> of ''M'' such that {{nowrap|''N''<sub>1</sub> ⊆ ''N''<sub>2</sub>}}, then the following two submodules are equal: {{nowrap|1=(''N''<sub>1</sub> + ''U'') ∩ ''N''<sub>2</sub> = ''N''<sub>1</sub> + (''U'' ∩ ''N''<sub>2</sub>)}}.

If ''M'' and ''N'' are left ''R''-modules, then a [[map (mathematics)|map]] {{nowrap|''f'' : ''M'' → ''N''}} is a '''[[module homomorphism|homomorphism of ''R''-modules]]''' if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'',
:<math>f(r \cdot m + s \cdot n) = r \cdot f(m) + s \cdot f(n)</math>.
This, like any [[homomorphism]] of mathematical objects, is just a mapping that preserves the structure of the objects.  Another name for a homomorphism of ''R''-modules is an ''R''-[[linear map]].

A [[bijective]] module homomorphism {{nowrap|''f'' : ''M'' → ''N''}} is called a module [[isomorphism]], and the two modules ''M'' and ''N'' are called '''isomorphic'''. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.

The [[kernel (algebra)|kernel]] of a module homomorphism {{nowrap|''f'' : ''M'' → ''N''}} is the submodule of ''M'' consisting of all elements that are sent to zero by ''f'', and the [[image (mathematics)|image]] of ''f'' is the submodule of ''N'' consisting of values ''f''(''m'') for all elements ''m'' of ''M''.<ref>{{Cite web|url=https://faculty.math.illinois.edu/~r-ash/Algebra/Chapter4.pdf|title=Module Fundamentals|last=Ash|first=Robert|website=Abstract Algebra: The Basic Graduate Year}}</ref> The [[isomorphism theorem]]s familiar from groups and vector spaces are also valid for ''R''-modules.

Given a ring ''R'', the set of all left ''R''-modules together with their module homomorphisms forms an [[abelian category]], denoted by ''R''-'''Mod''' (see [[category of modules]]).