What is the characteristic of the polynomial ring $\mathbb{Z}_6[x]$?

Question

What is the characteristic of the polynomial ring $\mathbb{Z}_6[x]$?

I want to clarify if my understanding is correct.

First, $\mathbb{Z}_6[x] =$ {$a_0 + a_1x + \cdots + a_nx^n$} where $a_i \in \mathbb{Z}_6$

So $a_i$ is one of the elements in the set: {$0,1,2,3,4,5$}

Now, to find characteristic, I need to look for $n$ where $n*a = 0$ for all $a \in \mathbb{Z}_6[x]$

Whatever polynomials I have in the ring, if I multiply a polynomial by 6, then all the coefficients of the polynomial function will go to $0$ since I do "mod 6" on each coefficient.

So 6 is the characteristic of the ring.

Thanks for reviewing!

Answer 1

It is almost correct. However, the characteristic of a commutative ring $R$ is the smallest $n\in\Bbb N$ such that $(\forall r\in R):nr=0$ (if such a $n$ exists; otherwise, the characteristic is $0$). So, you have to prove that if $k\in\{1,2,3,4,5\}$, then there is some $p(x)\in\Bbb Z_6[x]$ such that $kp(x)\ne0$. That's easy: take $p(x)=1$.