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Definition: (Unit) Ring

A ring is an algebraic structure \(R\) with two binary operations \( + \) and \(\cdot\), denoted by \((R, + ,\cdot)\), for which the following holds:

1. \((R, + )\) is an Abelian group,
2. \((R,\cdot)\) is a semigroup (i.e. the operation "$\cdot$" is associative),
3. The distributivity law holds for all \(x,y,z\in R\).

If \((R,\cdot)\) is a monoid (i.e. if the semigroup contains a multiplicative identity \(1\)), then the ring is called a unit ring (or ring with identity).

"Unfolding" all definitions, a 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 $x,y,z\in R.$
• 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 $x,y,z\in R.$

Mentioned in:

Chapters: 1 2 3 4
Definitions: 5 6 7 8 9 10 11 12 13 14 15 16
Lemmas: 17 18
Parts: 19
Proofs: 20 21 22 23 24 25 26 27 28 29 30
Propositions: 31 32 33 34 35 36 37 38
Sections: 39
Theorems: 40 41

Thank you to the contributors under CC BY-SA 4.0!

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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001

Adapted from CC BY-SA 3.0 Sources:

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