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]

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:

1. [math]\forall x,y\in R[xy=yx][/math]
2. [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

• Examples of rings

See also

• Group