Definition:Submodule

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ_G}_R$ be an $R$-module.

Let $H \subseteq G$ be a subset of $G$ which is closed for scalar product:

$\forall \lambda \in R, x \in H: \lambda \circ x \in H$

Let $\struct {H, +_H, \circ_H}_R$ be an $R$-module where:

$+_H$ is the restriction of $+_G$ to $H \times H$

$\circ_H$ is the restriction of $\circ_G$ to $R \times H$.

Then $\struct {H, +_H, \circ_H}_R$ is a submodule of $\struct {G, +_G, \circ_G}_R$.

Proper Submodule

Let $H$ be a proper subset of $G$.

Then $\struct {H, +_H, \circ_H}_R$ is a proper submodule of $\struct {G, +_G, \circ_G}_R$.

Also see

• Results about submodules can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases
• 2008: Steven Roman: Advanced Linear Algebra (3rd ed.): Chapter $4$: Modules I: Basic Properties: Submodules