Right Ideal is Right Module over Ring
Jump to navigation
Jump to search
Theorem
Let $\struct {R, +, \times}$ be a ring.
Let $J \subseteq R$ be a right ideal of $R$.
Let $\circ : J \times R \to J$ be the restriction of $\times$ to $J \times R$.
Then $\struct {J, +, \circ}$ is a right module over $\struct {R, +, \times}$.
Corollary
Let $\struct {R, +, \times}$ be a ring.
Then $\struct {R, +, \times}$ is a right module over $\struct {R, +, \times}$.
Proof
By definition of a right ideal then $\circ$ is well-defined.
Right Module Axiom $\text {RM} 1$: (Right) Distributivity over Module Addition
Follows directly from Ring Axiom $\text D$: Distributivity of Product over Addition.
$\Box$
Right Module Axiom $\text {RM} 2$: (Left) Distributivity over Scalar Addition
Follows directly from Ring Axiom $\text D$: Distributivity of Product over Addition.
$\Box$
Right Module Axiom $\text {RM} 3$: 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