information:extraction, structural:expansion
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 r ⋅ n (or 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 ⟨ X ⟩ = ⋂ N ⊇ X N {\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N} where N runs over the submodules of M that contain X, or explicitly { ∑ i = 1 k r i x i ∣ r i ∈ R, x i ∈ X } {\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}, which is important in the definition of tensor products of modules.

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 that satisfies the modular law:Given submodules U, N, N of M such that N ⊆ N, then the following two submodules are equal: (N + U) ∩ N = N + (U ∩ N).

If M and N are left R -modules, then a map f : M → N is a homomorphism of R -modules if for any m, n in M and r, s in R,

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 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 of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f (m) for all elements m of M. The isomorphism theorems 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).