Module theory MOC

Submodule

Let (𝑀,𝑅, +, ⋅) be a (left) module. A submodule 𝑁 ≤𝑀 is a module under the same operations, #m/def/module i.e. (𝑁, +) is a subgroup such that 𝑟 ⋅𝑛 ∈𝑁 for any 𝑛 ∈𝑁 and 𝑟 ∈𝑅. Thus a submodule is an invariant subspace under the carried representation of 𝑅 (see invariant subspace).

Examples

• Let 𝐼 ⊴𝑅 be an ideal. Then 𝐼 is an 𝑅-submodule of 𝑅.

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