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Let $R$ be a ring with identity. A left module $M$ over $R$ is a set with two binary operations, $+: M\times M \longrightarrow M$ and $\cdot: R \times M \longrightarrow M$ , such that
- $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in M$
- $\u+\v=\v+\u$ for all $\u,\v\in M$
- There exists an element $\0 \in M$ such that $\u+\0=\u$ for all $\u \in M$
- For any $\u \in M$ , there exists an element $\v \in M$ such that $\u+\v=\0$
- $a \cdot (b \cdot \u) = (a \cdot b) \cdot \u$ for all $a,b \in R$ and $\u \in M$
- $a \cdot (\u+\v) = (a \cdot \u) + (a \cdot \v)$ for all $a \in R$ and $\u,\v \in M$
- $(a + b) \cdot \u = (a \cdot \u) + (b \cdot \u)$ for all $a,b \in R$ and $\u \in M$
A left module $M$ over $R$ is called unitary or unital if $1_R \cdot \u = \u$ for all $\u \in M$ .
A (unitary or unital) right module is defined analogously, except that the function $\cdot$ goes from $M \times R$ to $M$ and the scalar multiplication operations act on the right. If $R$ is commutative, there is an equivalence of categories between the category of left $R$ -modules and the category of right $R$ -modules.
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"module" is owned by djao.
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Cross-references: category, equivalence of categories, commutative, right, act on, operations, multiplication, scalar, function, unital, unitary, binary operations, identity, ring
There are 100 references to this entry.
This is version 6 of module, born on 2001-10-19, modified 2004-11-19.
Object id is 365, canonical name is Module.
Accessed 19439 times total.
Classification:
| AMS MSC: | 13-00 (Commutative rings and algebras :: General reference works ) | | | 16-00 (Associative rings and algebras :: General reference works ) | | | 20-00 (Group theory and generalizations :: General reference works ) |
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Pending Errata and Addenda
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