R-module

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).



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As of this snapshot date, this entry was owned by bci1.