Difference between revisions of "Ring"
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==Definition== | ==Definition== | ||
| − | A set {{M|R}} and two [[Binary operation|binary operations]] {{M|+}} and {{M|\times}} such that the following hold: | + | A set {{M|R}} and two [[Binary operation|binary operations]] {{M|+}} and {{M|\times}} such that the following hold<ref>Fundamentals of abstract algebra - an expanded version - Neal H. McCoy</ref>: |
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| − | + | Is a ring, which we write: <math>(R,+:R\times R\rightarrow R,\times:R\times R\rightarrow R)</math> but because [[Mathematicians are lazy]] we write simply: | |
| + | * <math>(R,+,\times)</math> | ||
| − | == | + | ===Subring=== |
| − | {{ | + | If {{M|(S,+,\times)}} is a ring, and every element of {{M|S}} is also in {{M|R}} (for another ring {{M|(R,+,\times)}}) and the operations of addition and multiplication on {{M|S}} are the same as those on {{M|R}} (when restricted to {{M|S}} of course) then we say ''"{{M|S}} is a subring of {{M|R}}"'' |
| − | + | ||
| − | + | ||
| − | |||
| − | |||
| − | + | '''Note:'''<br/> | |
| − | + | 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== | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Name | ||
| + | ! Statement | ||
| + | ! Explanation | ||
| + | |- | ||
| + | ! Commutative Ring | ||
| + | | <math>\forall x,y\in R[xy=yx]</math> | ||
| + | | The order we multiply by does not matter. Calling a ring commutative isn't ambiguous because by definition addition in a ring is [[Commutative|commutative]] so when we call a ring commutative we must mean "it is a ring, and also multiplication is commutative". | ||
| + | |- | ||
| + | ! Ring with Unity | ||
| + | | <math>\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x]</math> | ||
| + | | The existence of a multiplicative identity, once we have proved it is unique we often denote this "{{M|1}}" | ||
| + | |} | ||
| + | ===Using properties=== | ||
| + | A ''commutative ring with unity'' is a ring with the additional properties of: | ||
| + | # <math>\forall x,y\in R[xy=yx]</math> | ||
| + | # <math>\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x]</math> | ||
| + | It is that simple. | ||
==Important theorem== | ==Important theorem== | ||
a0=0a=0 | a0=0a=0 | ||
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use a(a+0)=aa and go from there. | use a(a+0)=aa and go from there. | ||
| + | ==See next== | ||
| + | * [[Examples of rings]] | ||
| + | |||
| + | ==See also== | ||
| + | * [[Group]] | ||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Abstract Algebra}} | {{Definition|Abstract Algebra}} | ||
Revision as of 17:01, 19 May 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[1]:
| 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] |
Is a ring, which we write: [math](R,+:R\times R\rightarrow R,\times:R\times R\rightarrow R)[/math] but because Mathematicians are lazy we write simply:
- [math](R,+,\times)[/math]
Subring
If [ilmath](S,+,\times)[/ilmath] is a ring, and every element of [ilmath]S[/ilmath] is also in [ilmath]R[/ilmath] (for another ring [ilmath](R,+,\times)[/ilmath]) and the operations of addition and multiplication on [ilmath]S[/ilmath] are the same as those on [ilmath]R[/ilmath] (when restricted to [ilmath]S[/ilmath] of course) then we say "[ilmath]S[/ilmath] is a subring of [ilmath]R[/ilmath]"
Note:
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
| Name | Statement | Explanation |
|---|---|---|
| Commutative Ring | [math]\forall x,y\in R[xy=yx][/math] | The order we multiply by does not matter. Calling a ring commutative isn't ambiguous because by definition addition in a ring is commutative so when we call a ring commutative we must mean "it is a ring, and also multiplication is commutative". |
| Ring with Unity | [math]\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x][/math] | The existence of a multiplicative identity, once we have proved it is unique we often denote this "[ilmath]1[/ilmath]" |
Using properties
A commutative ring with unity is a ring with the additional properties of:
- [math]\forall x,y\in R[xy=yx][/math]
- [math]\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x][/math]
It is that simple.
Important theorem
a0=0a=0
use a(a+0)=aa and go from there.
See next
See also
References
- ↑ Fundamentals of abstract algebra - an expanded version - Neal H. McCoy
