Ring: Group + multiplication (see: http://en.wikipedia.org/wiki/Ring_%28mathematics%29)
and the three elements it defines:
additive identity aka zero
addition
multiplication
Note, if you have distributive property, additive inverses, and multiplicative identity you
can prove you have a commutative group under the ring:
(a + 1)*(b + 1) = a(b + 1) + (b + 1)
2. = ab + a + b + 1
3. or:
4.
5. = (a + 1)b + (a + 1)
6. = ab + b + a + 1
7.
8. So: ab + a + b + 1 == ab + b + a + 1
9. using the fact that -(ab) and -1 exist, we get:
10. a + b == b + a
Ring: Group + multiplication (see: http://en.wikipedia.org/wiki/Ring_%28mathematics%29) and the three elements it defines:
Note, if you have distributive property, additive inverses, and multiplicative identity you can prove you have a commutative group under the ring: