Definition: Commutative (Unit) Ring

A commutative ring is a ring $(R, +,\cdot)$, in which the binary operation "$\cdot$" is commutative.

A commutative unit ring ring is a commutative ring $(R, +,\cdot)$ with a neutral element of the operation "$\cdot$".

"Unfolding" all definitions, a commutative ring fulfills the following axioms:

Associativity of "$+$": $x+(y+z)=(x+y)+z$ for all $x,y,z\in R.$
Commutativity of "$+$": $x+y=y+x$ for all $x,y\in R.$
Neutral Element of "$+$": There is an element $0\in R$ with $0+x=x+0=x$ for all $x\in R.$
Inverse elements of "$+$": For all $x\in R$ there exists an $-x\in G$ with $x+(-x)=(-x)+x=0.$
Associativity of "$\cdot$": $x\cdot(y\cdot z)=(x\cdot y)\cdot z$ for all $x,y,z\in R.$
Commutativity of "$\cdot$": $x\cdot y=y\cdot x$ for all $x,y\in R.$
Neutral Element of "$\cdot$" (only when $R$ is a unit ring!): There is an element $1\in R$ with $1\cdot x=x\cdot 1=x$ for all $x\in R.$
Distributivity laws: $(x+y)\cdot z=x\cdot z + y\cdot z$ and $x\cdot (y+z)=x\cdot y + x\cdot z$ for all $x,y,z\in R.$