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R-module (Definition)

R-Module and left/right module definitions

Definition 0.1   Consider a ring $R$ with identity. Then a left module $M_L$ over $R$ is defined as a set with two binary operations,

$\displaystyle +: M_L \times M_L \longrightarrow M_L$
and

$\displaystyle \bullet : R \times M_L \longrightarrow M_L,$
such that
  1. $(\u +\v )+\mathbf{w}= \u +(\v +\mathbf{w})$ for all $\u ,\v ,\mathbf{w}\in M_L$
  2. $\u +\v =\v +\u$ for all $\u ,\v\in M_L$
  3. There exists an element $\mathbf{0} \in M_L$ such that $\u +\mathbf{0}=\u$ for all $\u\in M_L$
  4. For any $\u\in M_L$, there exists an element $\v\in M_L$ such that $\u +\v =\mathbf{0}$
  5. $a \bullet (b \bullet \u ) = (a \bullet b) \bullet \u$ for all $a,b \in R$ and $\u\in M_L$
  6. $a \bullet (\u +\v ) = (a \bullet\u ) + (a \bullet \v )$ for all $a \in R$ and $\u ,\v\in M_L$
  7. $(a + b) \bullet \u = (a \bullet \u ) + (b \bullet \u )$ for all $a,b \in R$ and $\u\in M_L$

A right module $M_R$ is analogously defined to $M_L$ except for two things that are different in its definition:

  1. the morphism “$\bullet$” goes from $M_R \times R$ to $M_R,$ and
  2. the scalar multiplication operations act on the right of the elements.
Definition 0.2   An R-module generalizes the concept of module to $n$-objects by employing Mitchell's definition of a “ring with n-objects” $R_n$; thus an $R$-module is in fact an $R_n$ module with this notation.

Remarks

One can define the categories of left- and - right R-modules, whose objects are, respectively, left- and - right R-modules, and whose arrows are R-module morphisms.

If the ring $R$ is commutative one can prove that the category of left $R$–modules and the category of right $R$–modules are equivalent (in the sense of an equivalence of categories, or categorical equivalence).



"R-module" is owned by bci1.

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Also defines:  module, R-module, ring with n-objects, category of left R-modules, categories of right R-modules
Keywords:  module, left- and - right R-modules, categories of left- and - right R-modules

Cross-references: objects, categories, concept, scalar, morphism, operations, identity
There are 19 references to this object.

This is version 10 of R-module, born on 2009-01-31, modified 2009-01-31.
Object id is 458, canonical name is RModule.
Accessed 1548 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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