# 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 $x,y,z\in R.$

Definitions: 1 2 3 4 5 6 7 8 9 10 11 12
Lemmas: 13 14 15 16 17
Proofs: 18 19 20 21
Propositions: 22 23 24

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück