Math 2.0

Associative rings and algebras Definition

Submodule

Builds on Module over a ring

Definition

Let RR be a ring with identity element and let MM be a left module over RR. A submodule of MM is a subset N⊆MN \subseteq M such that:

  1. NN is a subgroup of the abelian group (M,+)(M, +).
  2. rx∈Nrx \in N for all r∈Rr \in R and x∈Nx \in N.

Remarks

A submodule is a module over RR with the restricted operations. The submodules of RR, as a left module over itself, are the left ideals of RR. The ℤ\Z-submodules of an abelian group are its subgroups, and over a field the submodules are the subspaces. Intersections and sums of submodules are submodules.

Reading path

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Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Module_(mathematics), Submodules and homomorphisms
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