PlanetPhysics/R Module

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R-Module and left/right module definitionsEdit

Consider a ring   with identity. Then a left module   over   is defined as a set with two binary operations,   and   such that

  1. Failed to parse (unknown function "\u"): {\displaystyle (\u+\v)+\w = \u+(\v+\w)} for all Failed to parse (unknown function "\u"): {\displaystyle \u,\v,\w \in M_L}
  2. Failed to parse (unknown function "\u"): {\displaystyle \u+\v=\v+\u} for all Failed to parse (unknown function "\u"): {\displaystyle \u,\v\in M_L}
  3. There exists an element Failed to parse (syntax error): {\displaystyle \0 \in M_L} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \u+\0=\u} for all Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L}
  4. For any Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L} , there exists an element Failed to parse (unknown function "\v"): {\displaystyle \v \in M_L} such that Failed to parse (unknown function "\u"): {\displaystyle \u+\v=\0}
  5. Failed to parse (unknown function "\u"): {\displaystyle a \bullet (b \bullet \u) = (a \bullet b) \bullet \u} for all   and Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L}
  6. Failed to parse (unknown function "\u"): {\displaystyle a \bullet (\u+\v) = (a \bullet\u) + (a \bullet \v)} for all   and Failed to parse (unknown function "\u"): {\displaystyle \u,\v \in M_L}
  7. Failed to parse (unknown function "\u"): {\displaystyle (a + b) \bullet \u = (a \bullet \u) + (b \bullet \u)} for all   and Failed to parse (unknown function "\u"): {\displaystyle \u \in M_L}

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

  1. the morphism " " goes from   to   and
  1. the scalar multiplication operations act on the right of the elements.

An R-module generalizes the concept of module to  -objects by employing Mitchell's definition of a "ring with n-objects"  ; thus an  -module is in fact an   module with this notation.

RemarksEdit

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