Left Ideal is Left Module over Ring
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Theorem
Let $\struct {R, +, \times}$ be a ring.
Let $J \subseteq R$ be a left ideal of $R$.
Let $\circ : R \times J \to J$ be the restriction of $\times$ to $R \times J$.
Then $\struct {J, +, \circ}$ is a left module over $\struct {R, +, \times}$.
Corollary
Let $\struct {R, +, \times}$ be a ring.
Then $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$.
Proof
By definition of a left ideal then $\circ$ is well-defined.
Left Module Axiom $\text M 1$: (Left) Distributivity over Module Addition
Follows directly from Ring Axiom $\text D$: Distributivity of Product over Addition
$\Box$
Left Module Axiom $\text M 2$: (Right) Distributivity over Scalar Addition
Follows directly from Ring Axiom $\text D$: Distributivity of Product over Addition
$\Box$
Left Module Axiom $\text M 3$: Associativity
Follows directly from Ring Axiom $\text M1$: Associativity of Product
$\blacksquare$
Also see
Sources
- 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled