I've been solving some problems from my abstract algebra course as training for the final exam, and I want to check if my solution to this one is correct:
Describe the units, the nilpotent elements and the zero divisors of the rings $\mathbb{Z}_4[X]$ and $\mathbb{Z}_6[X].$
This is the work I did:
For $\mathbb{Z}_4[X]$:
- Units: this one's easy, since units in a polynomial ring $R[X]$ are the same as the units in $R$, so I conclude that the units of $\mathbb{Z}_4[X]$ are $\bar1,\bar3$ (the coprime elements to $4$).
- Zero divisors: First, it's obvious that $\bar 2$ is zero divisor. Then, in order for a polynomial to get cancelled by multiply6ing it by a non-zero element, it must have all of it's coefficients $\bar0$ or $\bar2$. So I conclude that the zero divisors of $\mathbb{Z}_4[X]$ are all the polynomials with $\bar 0$ and $\bar 2$ as coefficients (not including $\bar 0$).
- Nilpotent elements: in order for the power of a polynomial in $\mathbb{Z}_4[X]$ to be cancelled, the same reasoning from the zero divisors gives us that the nilpotent elements of $\mathbb{Z}_4[X]$ are the polynomials with coefficients $\bar0$ and $\bar 2$ (this time including $\bar 0$).
For $\mathbb{Z}_6[X]$:
- Units: The units in $\mathbb{Z}_6[X]$ are the same of $\mathbb{Z}_6$, so they are $\bar 1$ and $\bar 5$ (the coprimes to $6$)
- Zero divisors: using a similar reasoning to the one I used for $\mathbb{Z}_4[X]$, I end up concluding that the zero divisors in $\mathbb{Z}_6[X]$ are the polynomial with coefficients $\bar0$, $\bar2$, $\bar 4$; and the polynomials with coefficients $\bar0$, $\bar 3$ (not including $\bar 0$)(also not mixing coefficients, either it has $\bar2$, $\bar4$ or either it has $\bar3$).
- Nilpotent elements: in order for a power of a polynomial to be cancelled, it's clear that it's leading coefficient must be multiple of $6$, as well as it's independent coefficient, but $6$ is $\bar 0$ in $\mathbb{Z}_6[X]$, so I end up concluding that the only nilpotent element in $\mathbb{Z}_6[X]$ is $\bar 0$.
Is my solution correct? If not, where did I go wrong? Any help will be appreciated, thanks in advance.
