Left Ideal is Left Module over Ring/Ring is Left Module over Ring

Theorem

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

Then $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$.

Proof

From Ring is Ideal of Itself, $R$ is a left ideal.

From Left Ideal is Left Module over Ring, $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$.

$\blacksquare$

Also see

• Ring is Right Module over Ring

Sources

• 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled