Difference between revisions of "Ring"
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Some books introduce rings first, I do not know why. A ring is an additive [[Group|group]] (it is commutative making it an Abelian one at that), that is a ring is just a group {{M|(G,+)}} with another operation on {{M|G}} called {{M|\times}} | Some books introduce rings first, I do not know why. A ring is an additive [[Group|group]] (it is commutative making it an Abelian one at that), that is a ring is just a group {{M|(G,+)}} with another operation on {{M|G}} called {{M|\times}} | ||
| + | |||
| + | ==Properties== | ||
| + | {{Todo|I did these in a rush - just here for basic ref}} | ||
| + | ===Commutative ring=== | ||
| + | Multiplication is commutative | ||
| + | |||
| + | ===Ring with unity=== | ||
| + | There is a multiplicative identity | ||
| + | |||
| + | ==Multiplicative inverse== | ||
| + | For a ring with unity, if there exists an element s, such that as=sa=e then we call that the multiplicative inverse | ||
| + | |||
| + | ==Important theorem== | ||
| + | a0=0a=0 | ||
| + | |||
| + | use a(a+0)=aa and go from there. | ||
| + | |||
{{Definition|Abstract Algebra}} | {{Definition|Abstract Algebra}} | ||
Revision as of 13:04, 18 March 2015
Not to be confused with rings of sets which are a topic of algebras of sets and thus [ilmath]\sigma[/ilmath]-Algebras and [ilmath]\sigma[/ilmath]-rings
Contents
Definition
A set [ilmath]R[/ilmath] and two binary operations [ilmath]+[/ilmath] and [ilmath]\times[/ilmath] such that the following hold:
| Rule | Formal | Explanation |
|---|---|---|
| Addition is commutative | [math]\forall a,b\in R[a+b=b+a][/math] | It doesn't matter what order we add |
| Addition is associative | [math]\forall a,b,c\in R[(a+b)+c=a+(b+c)][/math] | Now writing [ilmath]a+b+c[/ilmath] isn't ambiguous |
| Additive identity | [math]\exists e\in R\forall x\in R[e+x=x+e=x][/math] | We do not prove it is unique (after which it is usually denoted 0), just "it exists" The "exists [ilmath]e[/ilmath] forall [ilmath]x\in R[/ilmath]" is important, there exists a single [ilmath]e[/ilmath] that always works |
| Additive inverse | [math]\forall x\in R\exists y\in R[x+y=y+x=e][/math] | We do not prove it is unique (after we do it is usually denoted [ilmath]-x[/ilmath], just that it exists The "forall [ilmath]x\in R[/ilmath] there exists" states that for a given [ilmath]x\in R[/ilmath] a y exists. Not a y exists for all [ilmath]x[/ilmath] |
| Multiplication is associative | [math]\forall a,b,c\in R[(ab)c=a(bc)][/math] | |
| Multiplication is distributive | [math]\forall a,b,c\in R[a(b+c)=ab+ac][/math] [math]\forall a,b,c\in R[(a+b)c = ac+bc][/math] |
Some books introduce rings first, I do not know why. A ring is an additive group (it is commutative making it an Abelian one at that), that is a ring is just a group [ilmath](G,+)[/ilmath] with another operation on [ilmath]G[/ilmath] called [ilmath]\times[/ilmath]
Properties
TODO: I did these in a rush - just here for basic ref
Commutative ring
Multiplication is commutative
Ring with unity
There is a multiplicative identity
Multiplicative inverse
For a ring with unity, if there exists an element s, such that as=sa=e then we call that the multiplicative inverse
Important theorem
a0=0a=0
use a(a+0)=aa and go from there.
