Difference between revisions of "Ring"
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use a(a+0)=aa and go from there. | use a(a+0)=aa and go from there. | ||
| + | ==Important Theorems== | ||
| + | {{Begin Theorem}} | ||
| + | Theorem: The additive identity of a ring {{M|R}} is unique (and as such can be denoted {{M|0}} unambiguously) | ||
| + | {{Begin Proof}} | ||
| + | This is a classic "suppose there are two" proof, and we will do the same. | ||
| + | |||
| + | Suppose that {{M|0\in R}} is such that {{M|1=\forall x\in R[0+x=x+0=x]}} | ||
| + | : Suppose that {{M|0'\in R}} with {{M|1=0'\ne 0}} and also such that: {{M|1=\forall x\in R[0'+x=x+0'=x]}} | ||
| + | |||
| + | We will show that {{M|1=0=0'}}, contradicting them being different! Thus showing there is no other "zero" | ||
| + | |||
| + | '''Proof:''' | ||
| + | : <math>0+0'=0</math> by the property of {{M|0}} | ||
| + | : <math>0+0'=0'+0</math> by the commutivity of addition | ||
| + | :: <math>0'+0=0'</math> by the property of {{M|0'}} | ||
| + | : Thus <math>0=0'</math> | ||
| + | :: This contradicts that {{M|0\ne 0'}} so the claim they are distinct cannot be, we have only one "zero element", which herein we shall denote as "{{M|0}}" | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | {{Begin Theorem}} | ||
| + | Theorem: if {{M|1=a+c=b+c}} then {{M|1=a=b}} (and due to commutivity of addition <math>c+a=c+b\implies a=b</math> too) | ||
| + | {{Begin Proof}} | ||
| + | Suppose that {{M|1=a+c=b+c}} | ||
| + | : By the ''additive inverse'' property, <math>\exists x\in R:c+x=0</math> | ||
| + | :: First notice that <math>(a+c)+x=(b+c)+x</math> (using <math>a+c=b+c</math>) | ||
| + | ::* Let us take <math>(a+c)+x</math> | ||
| + | ::*: By associativity of addition, <math>(a+c)+x=a+(c+x)=a+0=a</math> | ||
| + | ::* Let us take <math>(b+c)+x</math> | ||
| + | ::*: By associativity of addition, <math>(b+c)+x=b+(c+x)=b+0=b</math> | ||
| + | :: We see that <math>a=a+c+x=b+c+x=b</math> | ||
| + | : Which is indeed just <math>a=b</math> | ||
| + | |||
| + | As claimed. | ||
| + | |||
| + | |||
| + | |||
| + | '''Note:''' | ||
| + | : Note that <math>c+a=b+c\implies a=b</math>, this can be proved identically to the above (but adding x to the left) or by: | ||
| + | :: <math>c+a=a+c</math> and </math>b+c=c+b</math> and then apply the above. | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | {{Begin Theorem}} | ||
| + | Theorem: The additive inverse of an element is unique (and herein, for a given {{M|x\in R}} shall be denoted {{M|-x}}) | ||
| + | {{Begin Proof}} | ||
| + | {{Todo}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
==See next== | ==See next== | ||
* [[Examples of rings]] | * [[Examples of rings]] | ||
Revision as of 06:43, 21 May 2015
Not to be confused with rings of sets which are a topic of algebras of sets and thus σ-Algebras and σ-rings
Contents
[hide]Definition
A set R and two binary operations + and × such that the following hold[1]:
| Rule | Formal | Explanation |
|---|---|---|
| Addition is commutative | ∀a,b∈R[a+b=b+a] |
It doesn't matter what order we add |
| Addition is associative | ∀a,b,c∈R[(a+b)+c=a+(b+c)] |
Now writing a+b+c isn't ambiguous |
| Additive identity | ∃e∈R∀x∈R[e+x=x+e=x] |
We do not prove it is unique (after which it is usually denoted 0), just "it exists" The "exists e forall x∈R" is important, there exists a single e that always works |
| Additive inverse | ∀x∈R∃y∈R[x+y=y+x=e] |
We do not prove it is unique (after we do it is usually denoted −x, just that it exists The "forall x∈R there exists" states that for a given x∈R a y exists. Not a y exists for all x |
| Multiplication is associative | ∀a,b,c∈R[(ab)c=a(bc)] |
|
| Multiplication is distributive | ∀a,b,c∈R[a(b+c)=ab+ac] ∀a,b,c∈R[(a+b)c=ac+bc] |
Is a ring, which we write: (R,+:R×R→R,×:R×R→R)
- (R,+,×)
Subring
If (S,+,×) is a ring, and every element of S is also in R (for another ring (R,+,×)) and the operations of addition and multiplication on S are the same as those on R (when restricted to S of course) then we say "S is a subring of R"
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 (G,+) with another operation on G called ×
Properties
| Name | Statement | Explanation |
|---|---|---|
| Commutative Ring | ∀x,y∈R[xy=yx] |
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 | ∃e×∈R∀x∈R[xe×=e×x=x] |
The existence of a multiplicative identity, once we have proved it is unique we often denote this "1" |
Using properties
A commutative ring with unity is a ring with the additional properties of:
- ∀x,y∈R[xy=yx]
- ∃e×∈R∀x∈R[xe×=e×x=x]
It is that simple.
Important theorem
a0=0a=0
use a(a+0)=aa and go from there.
Important Theorems
Theorem: The additive identity of a ring R is unique (and as such can be denoted 0 unambiguously)
Theorem: if a+c=b+c then a=b (and due to commutivity of addition c+a=c+b⟹a=b
Theorem: The additive inverse of an element is unique (and herein, for a given x∈R shall be denoted −x)
See next
See also
References
- Jump up ↑ Fundamentals of abstract algebra - an expanded version - Neal H. McCoy
