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Unit Ring

A unit ring is a ring with a multiplicative identity. It is therefore sometimes also known as a "ring with identity."

It is given by a set together with two binary operators S(+,*) satisfying the following conditions:

1. Additive associativity: For all a,b,c in S, (a+b)+c=a+(b+c),

2. Additive commutativity: For all a,b in S, a+b=b+a,

3. Additive identity: There exists an element 0 in S such that for all a in S:0+a=a+0=a,

4. Additive inverse: For every a in S, there exists a -a in S such that a+(-a)=(-a)+a=0,

5. Multiplicative associativity: For all a,b,c in S, (a*b)*c=a*(b*c),

6. Multiplicative identity: There exists an element 1 in S such that for all a in S, 1*a=a*1=a,

7. Left and right distributivity: For all a,b,c in S, a*(b+c)=(a*b)+(a*c) and (b+c)*a=(b*a)+(c*a).

See also

Binary Operator, Ring

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References

Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.

Referenced on Wolfram|Alpha

Unit Ring

Cite this as:

Weisstein, Eric W. "Unit Ring." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnitRing.html

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