So my class on ring theory recently began, and I'm having a bit of trouble understanding ideals that are generated by polynomials. For an arbitrary ring, I know the definition of such an ideal, that is, for $R$ a ring and $a \in R$, $(a) = \{ \sum_{i=1}^{n} r_ias_i \mid r_i,s_i \in R \}$. So, then, this set is essentially just all the possible sums of different combinations of $r_ias_i$? Say, for $n=1$, $r_1as_1 \in (a)$, where $r_1,s_1$ just "run through" all the elements in $R$?
Specifically, I had a previous homework problem concerning the following ring of polynomials:
$R = [\mathbb{Z}/2\mathbb{Z}](t)$. I know what this means as a set, but I had trouble understanding this ideal: $f = f(t) = t^2 + t + 1 \in R$ and the ideal being $(f)$. The question was regarding $R/(f)$, which I know is the set $\{r + (f) \mid r \in R \}$, but I didn't even know where to start due to my lack of understanding of $(f)$. Any help would in understanding would be welcomed
Best Answer
Put in simple words,
I give here just a heuristic:
I like to think of the quotient ring $R/(a)$ as having the same elements of $R$ but the equality is modified so that all elements of $(a)$ be equal to zero in the quotient. Observe that it is enough to require $a=0$, because $ras=0$ and $\sum r_ias_i=0$ already follows by the properties of equality.
In the case of polynomial ring, a factor like $R[t]/(f)$ with, say $f=t^n-a_{n-1}t^{n-1}-\dots-a_1t-a_0$ will always be represented by the set of polynomials of degree $<n$, because in the quotient ring we have $$t^n=a_{n-1}t^{n-1}+\dots+a_1t+a_0$$ So that, any time a $t^n$ appears, it can be replaced by a degree $<n$ polynomial in $R[t]/(f)$.