Difference between revisions of "Module"
From Maths
m |
m (→Definition: Typo) |
||
| Line 4: | Line 4: | ||
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]}}, | ||
| Line 14: | Line 14: | ||
</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}} | ||
Latest revision as of 22:40, 19 October 2016
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Important for the Rings and Modules. Demote when fleshed out
Contents
[hide]Definition
Let (R,+,∗,0)[Note 1] be a ring - not necessarily with unity - then a "left R-module"[1] is:
- An Abelian group, (M,⊕) together with a
- left action, [:R×M→M] given by [:(r,x)↦rx] of R on M, called the "left R-module structure" on M
such that:
- ∀r,s∈R,∀x∈M[r(sx)=(rs)x],
- ∀r,s∈R,∀x∈M[(r+s)x=rx+sx] and
- ∀r∈R,∀x,y∈M[r(x+y)=rx+ry]
Additionally, if R is a u-ring[Note 2] then a left R-module is unital when[1]:
- ∀x∈M[1Rx=x]
The notation RM generally indicates that M is a left R-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
- Jump up ↑ Or (R,+,∗,0,1) if the ring has unity. Standard notation
- Jump up ↑ has unity, a multiplicative identity denoted 1 or 1R
