Is my interpretation of the 'polynomial quotient ring' $\mathbb{Z}_p [x] / f(x)$ correct?

Question

Within the book 'Introductory Lectures on Rings and Modules' (by John A. Beachy) the following definition is given:

If $R$ is a ring and $I$ is an ideal of $R$ then $R / I$ is the set of cosets $\{ a + I \mid a \in R \}$ of $I$ in $R$.

Based on this definition, I have been trying to interpret the meaning of the 'polynomial quotient ring' $\mathbb{Z}_p [x] / f(x)$, which I have interpreted to be the set of cosets of $f(x)$ in $\mathbb{Z}_p [x]$. That is $$ \mathbb{Z}_p [x] / f(x) = \{ f(x) + g(x) \mid g(x) \in \mathbb{Z}_p [x] \} $$

Is this correct?

Answer 1

The quotient is really done by the ideal generated by $f$. These are all the polynomials that are multiples of $f$.

You can see the elements of the quotient as the sets $g(x)+f(x)\mathbb{Z}_p[x]$ for $g\in\mathbb{Z}_p[x]$.

If you use Euclid's algorithm you can choose $g$ to have degree strictly smaller than $f$ by replacing it with the remainder of the division of $g$ by $f$. With that choice the representation of the elements of the quotient is unique.

Answer 2

What you wrote is not correct. First of all, note that the set you are referring to is equal to $\mathbb{Z}_p[x]$. You can see that $\{f(x) + g(x)\: | \: g(x)\in\mathbb{Z}_p[x]\}$ is a subset of $\mathbb{Z}_p[x]$. It is actually equal to it because for any element $a(x)\in\mathbb{Z}_p[x]$ we can choose $g(x)=a(x)-f(x)$.

A quotient ring, as defined above, is not a subset of the original ring, but the subset of its power set. In other words, every element of the quotient ring is a subset of the original ring. These subsets are of the following type - they are $a+I$ where $I$ is the ideal by which we quotient, and $a$ an element of the ring.

For example if $R=\mathbb{Z}$, $I=7\mathbb{Z}=\{\ldots,-14,-7,0,7,14,\ldots\}$, one of the elements of the quotient ring $\mathbb{Z}/7\mathbb{Z}$ is the following (with $a=4$) $$ 4+7\mathbb{Z}=\{4+x\:|\:x\in\mathbb{Z}\}=\{\ldots,-10,-3,4,11,18,\ldots\}. $$

The whole quotient ring $\mathbb{Z}/7\mathbb{Z}$ is the union of all the subsets of the above type, running over all $a\in\mathbb{Z}$, so $$ \mathbb{Z}/7\mathbb{Z}=\{ \{a+x\:|\:x\in7\mathbb{Z}\} \: | \: a\in\mathbb{Z} \}. $$ Note that even though the $a$'s are infinite, the quotient ring is finite, because some sets overlap.

In the polynomial ring case, note that you are not doing the quotient by a polynomial, but by the ideal generated by the polynomial. So your $I$ will be $$ I = (f(x))=f(x)\mathbb{Z}_p[x]=\{ f(x)h(x)\:|\:h(x)\in \mathbb{Z}_p[x]\} $$ I think you can take it up from here.