Those properties define a ring. (That's why I asked for the definition.) Now you need to show that the thing they've given you fulfills the definition; that is, that the elements of that thing obey these properties.
For instance, given [a]7, 7, and [c]7, can you should that ([a]7 + 7) + [c]7 = [a]7 + (7 + [c]7)?
ok, I've figured out all of those and proved them.
Now I'm working on showing Z7 is an integral domain....i have 2 choices, and I'm not sure if either is correct.
Choice 1:
Assume a and b are in z7
Then ab = 0 iff a = 0 or b = 0.
Suppose a and b do not equal 0 in z7
Since 7 is prime, neither a nor b will divide 7
Since 7 is prime, and neither a nor b will divide 7, then ab will also not divide 7.
Thus ab does not equal and there are no zero divisors.
Therefore, z7 is an integral domain.
Choice 2:
Since n = 7 is prime it can only be divided by 1 or 7.
For all a values in z7, a must be less than 7. Therefore, the divisors must also be less than 7.
Gcf (a, 7) = 1 because a must be less than 7 and 7 can only be divided by 1 and 7.
Therefore, z7is an integral domain because there are no zero divisors.
Thanks again for the help.