Ring

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

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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.