Associative rings and algebras Definition

Module over a ring

Definition

Let $R$ be a ring with identity element $1$ . A left module over $R$ is an abelian group $(M, +)$ together with a map $R \times M \to M$ , $(r, x) \mapsto rx$ , such that, for all $r, s \in R$ and $x, y \in M$ :

$r(x + y) = rx + ry$ .
$(r + s)x = rx + sx$ .
$(rs)x = r(sx)$ .
$1x = x$ .

Remarks

If $R$ is a field, then an $R$ -module is the same as a vector space over $R$ . Each abelian group is a module over the ring $\Z$ , with $nx = x + \cdots + x$ ( $n$ terms) for $n > 0$ . The ring $R$ is a module over itself, and $R^n$ is a module with the operations on coordinates. Right modules use a map $M \times R \to M$ ; over a commutative ring, the two notions are the same.

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Tag	`09VDS2W1E8`
URI	`m2://09VDS2W1E8`
Kind	definition
Writing	W1 stated
MSC	16D10, 13C05
Defines	Module over a ring

Module over a ring

Definition

Remarks

Where this leads

Further reading

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