Unit Ring

A unit ring is 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 $(b+c)*a=(b*a)+(c*a)$.

Thus, a unit ring is a Ring with a multiplicative identity.

See also Binary Operator, Ring

References

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