I'm confused on what units and non-trivial divisors of zero are when it comes to rings. For example, say I have this finite ring: R=GF(2)[x] mod x^3 + 1 = 0.
Now I know the elements are 0, 1, x, x + 1, x^2, x^2 + 1, x^2 + x, and x^2 + x + 1.
Aren't those all non-trivial divisors of zero besides 1?
And for the units, I read a unit of a ring is one of those elements, we'll say 'e', such that there exists the inverse of 'e' where e * e^-1 = 1. Do I multiply each of these elements by its inverse to find the units?
Also, does it matter if we can't obtain a field or not to find these two things? I know the example above doesn't give a field, but something like x^3 + x + 1, which has the same elements, does give a field.
Best Answer
In a field, all elements other than $0$ are units, and there are no nontrivial zero-divisors. In a finite ring (with 1), any non-unit is a zero-divisor. That's not true in general for infinite rings.