Ring

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

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

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TODO: I did these in a rush - just here for basic ref

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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.