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.
A unit in a ring is an element which has an inverse (i.e., an $a$ such that there is $b$ such that $a \cdot b = 1$). A zero divisor is an element $a \ne 0$ such that there is a $b \ne 0$ with $a \cdot b = 0$ (here I use the names $1$ for the multiplicative identity and $0$ for the additive one).
The elements of your ring are rather few; you can work out the full multiplication table and check for the above. Bonus is that your ring is commutative, as is easy to prove.