What does $\Bbb Z/2 \Bbb Z[X]$ and $\Bbb Z/3 \Bbb Z[X]$ mean? (abstract algebra)

Question

Can someone tell what does this mean: $\Bbb Z/2 \Bbb Z[X]$ and $\Bbb Z/3 \Bbb Z[X]$

I understand that $\Bbb Z/2 \Bbb Z$ means the integers numbers mod $2$, but I don't understand the last part. I know that $\Bbb Z[x]$ means a polynomial, but how can I take mod to a polynomial?

Answer 1

$\mathbb{Z}/n\mathbb{Z}[x]$ means the set(ring) of all polynomials $P(x)$ such that the coefficients are from $\mathbb{Z}/n\mathbb{Z}$. i.e., $$ \mathbb{Z}/n\mathbb{Z}[x] := \{ P(x) = a_nx^n + a_{n - 1}x^{n -1} + \cdots a_0 : a_i \in \mathbb{Z}/n\mathbb{Z} \} $$ E.G: $~x^2 + 2x + 1$ in $\mathbb{Z}/2\mathbb{Z} [x]$ is $x^2 + 1$.

Answer 2

It should be rather $(\mathbb Z/2\mathbb Z)[X]$ which is the ring of polynomials with coefficient in $\mathbb Z/2\mathbb Z$, i.e. for example $$1+X+X^2\quad \text{or}\quad X+X^8.$$

Same for $(\mathbb Z/3\mathbb Z)[X]$, it's the ring of polynomials with coefficients in $\mathbb Z/3\mathbb Z$.

Answer 3

For any ring, $\mathcal R$, you can show that $\mathcal R[x]:=\{\sum_{i=0}^nr_ix^i: r_i\in \mathcal R\}$, the set of polynomials in $x$ with coefficients in $\mathcal R$, also forms a ring, with polynomial addition and multiplication as the operations.

Now just set $\mathcal R=\Bbb Z/n\Bbb Z$. You get the ring of polynomials all of whose coefficients are in $\{0,1,2,\dots,n-1\}$.