Definition:Submodule/Proper

Definition

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

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

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

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

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