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Important for the Rings and Modules. Demote when fleshed out


Let [ilmath](R,+,*,0)[/ilmath][Note 1] be a ring - not necessarily with unity - then a "left [ilmath]R[/ilmath]-module"[1] is:

  • An Abelian group, [ilmath](M,\oplus)[/ilmath] together with a
  • left action, [ilmath][:R\times M\rightarrow M][/ilmath] given by [ilmath][:(r,x)\mapsto rx][/ilmath] of [ilmath]R[/ilmath] on [ilmath]M[/ilmath], called the "left [ilmath]R[/ilmath]-module structure" on [ilmath]M[/ilmath]

such that:

  1. [ilmath]\forall r,s\in R,\forall x\in M[r(sx)=(rs)x][/ilmath],
  2. [ilmath]\forall r,s\in R,\forall x\in M[(r+s)x=rx+sx][/ilmath] and
  3. [ilmath]\forall r\in R,\forall x,y\in M[r(x+y)=rx+ry][/ilmath]

Additionally, if [ilmath]R[/ilmath] is a u-ring[Note 2] then a left [ilmath]R[/ilmath]-module is unital when[1]:

  1. [ilmath]\forall x\in M[1_Rx=x][/ilmath]

The notation [ilmath]{}_RM[/ilmath] generally indicates that [ilmath]M[/ilmath] is a left [ilmath]R[/ilmath]-module

See next


  1. Or [ilmath](R,+,*,0,1)[/ilmath] if the ring has unity. Standard notation
  2. has unity, a multiplicative identity denoted [ilmath]1[/ilmath] or [ilmath]1_R[/ilmath]


  1. 1.0 1.1 Abstract Algebra - Pierre Antoine Grillet