Submodule Test
Theorem
Let $\struct {R, +_R, \times_R}$ be a ring with unity $1_R$.
Let $\struct {G, +, \circ}_R$ be a unitary $R$-module.
Let $H$ be a non-empty subset of $G$.
Then $\struct {H, +, \circ}_R$ is a submodule of $G$ if and only if:
| \(\text {(SM1)}: \quad\) | \(\ds \forall x, y \in H: \, \) | \(\ds x + y\) | \(\in\) | \(\ds H\) | ||||||||||
| \(\text {(SM2)}: \quad\) | \(\ds \forall x \in H: \forall \lambda \in R: \, \) | \(\ds \lambda \circ x\) | \(\in\) | \(\ds H\) |
Proof
Necessary Condition
Let $\struct {H, +, \circ}_R$ fulfil the conditions $\text {SM} 1$ and $\text {SM} 2$.
We have by hypothesis that $H \subseteq G$ such that $H \ne \O$.
We have from $\text {SM} 1$:
- $\forall x, y \in H: x + y \in H$
We have by hypothesis that $\struct {R, +_R, \times_R}$ is a ring with unity whose unity is $1_R$.
Hence:
| \(\ds x\) | \(\in\) | \(\ds H\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-1_R} \circ x\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -x\) | \(\in\) | \(\ds H\) | Definition of Unity of Ring |
It follows from the Two-Step Subgroup Test that $H$ is a subgroup of $\struct {G, +}$.
It remains to be shown that $\struct {H, +, \circ}_R$ fulfils the module axioms as follows:
| \((\text M 1)\) | $:$ | Scalar Multiplication (Left) Distributes over Module Addition | \(\ds \forall \lambda \in R: \forall x, y \in H:\) | \(\ds \lambda \circ \paren {x + y} \) | \(\ds = \) | \(\ds \paren {\lambda \circ x} + \paren {\lambda \circ y} \) | |||
| \((\text M 2)\) | $:$ | Scalar Multiplication (Right) Distributes over Scalar Addition | \(\ds \forall \lambda, \mu \in R: \forall x \in H:\) | \(\ds \paren {\lambda +_R \mu} \circ x \) | \(\ds = \) | \(\ds \paren {\lambda \circ x} + \paren {\mu \circ x} \) | |||
| \((\text M 3)\) | $:$ | Associativity of Scalar Multiplication | \(\ds \forall \lambda, \mu \in R: \forall x \in H:\) | \(\ds \paren {\lambda \times_R \mu} \circ x \) | \(\ds = \) | \(\ds \lambda \circ \paren {\mu \circ x} \) |
| \(\ds \forall x, y \in H: \, \) | \(\ds x + y\) | \(\in\) | \(\ds H\) | from $\text {SM} 1$ | ||||||||||
| \(\ds \lambda \circ x\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$ | |||||||||||
| \(\ds \lambda \circ y\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}\) | \(\in\) | \(\ds H\) | from $\text {SM} 1$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}\) | \(\in\) | \(\ds G\) | as $H \subseteq G$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}\) | \(=\) | \(\ds \lambda \circ \paren {x + y}\) | Module Axiom $\text M 1$: Distributivity over Module Addition as applied to $\struct {G, +, \circ}_R$ |
Hence $\struct {H, +, \circ}_R$ fulfils Module Axiom $\text M 1$: Distributivity over Module Addition.
| \(\ds \forall x \in H: \forall \lambda \in R: \, \) | \(\ds \lambda \circ x\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall \mu \in R: \, \) | \(\ds \paren {\lambda +_R \mu} \circ x\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$: as $\lambda +_R \mu \in R$ | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda +_R \mu} \circ x\) | \(\in\) | \(\ds G\) | as $H \subseteq G$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda +_R \mu} \circ x\) | \(=\) | \(\ds \paren {\lambda \circ x} + \paren {\mu \circ x}\) | Module Axiom $\text M 2$: Distributivity over Scalar Addition as applied to $\struct {G, +, \circ}_R$ |
Hence $\struct {H, +, \circ}_R$ fulfils Module Axiom $\text M 2$: Distributivity over Scalar Addition.
| \(\ds \forall x \in H: \forall \lambda \in R: \, \) | \(\ds \lambda \circ x\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall \mu \in R: \, \) | \(\ds \paren {\lambda \times_R \mu} \circ x\) | \(\in\) | \(\ds H\) | from $\text {SM} 2$: as $\lambda \times_R \mu \in R$ | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda \times_R \mu} \circ x\) | \(\in\) | \(\ds G\) | as $H \subseteq G$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\lambda \times_R \mu} \circ x\) | \(=\) | \(\ds \lambda \circ \paren {\mu \circ x}\) | Module Axiom $\text M 3$: Associativity as applied to $\struct {G, +, \circ}_R$ |
Hence $\struct {H, +, \circ}_R$ fulfils Module Axiom $\text M 3$: Associativity.
Sufficient Condition
Let $\struct {H, +, \circ}_R$ be a submodule of $G$.
As $H$ is an $R$-module, $\struct {H, +}$ is an abelian group.
As $\struct {H, +}$ is a group, it is closed under $+$ it follows that
- $\forall x, y \in H: x + y \in H$
which is $\text {SM} 1$.
As $H$ is an $R$-module, it is closed for scalar product:
- $\forall \lambda \in R, x \in H: \lambda \circ x \in H$
which is $\text {SM} 2$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Theorem $27.1$