Difference between revisions of "Module"
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Let {{M|(R,+,*,0)}}<ref group="Note">Or {{M|(R,+,*,0,1)}} if the ring has unity. Standard notation</ref> be a [[ring]] - not necessarily with [[ring with unity|unity]] - then a "''left'' {{M|R}}-module"{{rAAPAG}} is: | Let {{M|(R,+,*,0)}}<ref group="Note">Or {{M|(R,+,*,0,1)}} if the ring has unity. Standard notation</ref> be a [[ring]] - not necessarily with [[ring with unity|unity]] - then a "''left'' {{M|R}}-module"{{rAAPAG}} is: | ||
* An [[Abelian group]], {{M|(M,\oplus)}} together with a | * An [[Abelian group]], {{M|(M,\oplus)}} together with a | ||
| − | * left action, {{M|[:R\times M\rightarrow M]}} given by {{M|[:(r,x)\mapsto rx]}} of {{M|R}} on {{M| | + | * left action, {{M|[:R\times M\rightarrow M]}} given by {{M|[:(r,x)\mapsto rx]}} of {{M|R}} on {{M|M}}, called the "left {{M|R}}-module structure" on {{M|M}} |
such that: | such that: | ||
# {{M|1=\forall r,s\in R,\forall x\in M[r(sx)=(rs)x]}}, | # {{M|1=\forall r,s\in R,\forall x\in M[r(sx)=(rs)x]}}, | ||
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</ol> | </ol> | ||
The notation {{M|{}_RM}} generally indicates that {{M|M}} is a ''left'' {{M|R}}-module | The notation {{M|{}_RM}} generally indicates that {{M|M}} is a ''left'' {{M|R}}-module | ||
| + | |||
==See next== | ==See next== | ||
* [[Direct product of modules]] - an instance of a {{link|product|category theory}} | * [[Direct product of modules]] - an instance of a {{link|product|category theory}} | ||
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Important for the Rings and Modules. Demote when fleshed out
Contents
Definition
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:
- [ilmath]\forall r,s\in R,\forall x\in M[r(sx)=(rs)x][/ilmath],
- [ilmath]\forall r,s\in R,\forall x\in M[(r+s)x=rx+sx][/ilmath] and
- [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]:
- [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
- Direct product of modules - an instance of a product
- External direct sum of modules - an instance of a co-product
- Homomorphism
Notes
- ↑ Or [ilmath](R,+,*,0,1)[/ilmath] if the ring has unity. Standard notation
- ↑ has unity, a multiplicative identity denoted [ilmath]1[/ilmath] or [ilmath]1_R[/ilmath]
