Difference between revisions of "Ring"

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(Created page with "Not to be confused with rings of sets which are a topic of algebras of sets and thus Algebras}} and Sigma-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}}
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==Properties==
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{{Todo|I did these in a rush - just here for basic ref}}
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===Commutative ring===
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Multiplication is commutative
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===Ring with unity===
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There is a multiplicative identity
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==Multiplicative inverse==
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For a ring with unity, if there exists an element s, such that as=sa=e then we call that the multiplicative inverse
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==Important theorem==
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a0=0a=0
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use a(a+0)=aa and go from there.
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{{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 \sigma-Algebras and \sigma-rings


Definition

A set R and two binary operations + and \times such that the following hold:

Rule Formal Explanation
Addition is commutative \forall a,b\in R[a+b=b+a] It doesn't matter what order we add
Addition is associative \forall a,b,c\in R[(a+b)+c=a+(b+c)] Now writing a+b+c isn't ambiguous
Additive identity \exists e\in R\forall x\in 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\in R" is important, there exists a single e that always works

Additive inverse \forall x\in R\exists y\in 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\in R there exists" states that for a given x\in R a y exists. Not a y exists for all x

Multiplication is associative \forall a,b,c\in R[(ab)c=a(bc)]
Multiplication is distributive \forall a,b,c\in R[a(b+c)=ab+ac]

\forall a,b,c\in R[(a+b)c = ac+bc]

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 \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.