Untermodul

Untermodul is a submodule of an R-Modul M, where R is a ring (often with identity). A Untermodul N ⊆ M is an additive Untergruppe of M that is closed under the action of R: r · n ∈ N for all r ∈ R and n ∈ N. This makes N itself an R-Modul with the restricted operation.

Examples help illustrate the concept. If V is a vector space over a field F, then every Untermodul of V is simply a Untergruppe that is closed under scalar multiplication by F, i.e., a subspace. For the ring R = Z and M = Z (as a Z-Modul), the Untermoduln of M are exactly the subgroups nZ for integers n ≥ 0.

Generated Untermodul and submodule operations. For a subset S ⊆ M, the generated Untermodul ⟨S⟩ is the smallest submodul of M containing S. Submodules can be combined by sum and intersection, both of which are again submodulów. The lattice of submodules of M forms a structure that reflects the internal composition of M.

Quotients and homomorphisms. If N ≤ M, the quotient M/N is a module with cosets, and the natural projection π: M → M/N is a module homomorphism. Kernels of module homomorphisms are submodulów of the domain, and the correspondence theorem relates submodules of quotients to submodules of the original module containing N.

Maximal and simple submodules; structure theory. A submodul N of M is maximal if it is proper and not contained in any larger proper submodul. A module is simple if it has no nontrivial submodulów. Semisimple modules decompose as direct sums of simple submodules, a cornerstone of module theory.