Describe $\mathbb{Q}[x] / (x^2 + x)$ and $\mathbb{Z}[x]/ (x-2, x^2+1)$ in simpler terms. I am trying to review for a test. One of the questions is asking me to find a simpler description of these two rings and I am not sure what they mean by this. Any suggestions?
 A: I find the easiest setting to think about questions like these are by using the first isomorphism theorem: if $\phi$ is a ring homomorphism from $R$ onto $S$, then $S \simeq R/\ker(\phi)$. This means you have to have some suspicion about what the result will look like, define a homomorphism into that ring, show that it is onto, and then prove that the kernel is what you expected. (This last step is usually done in two parts, establishing $I = \ker(\phi)$ via $I \subseteq \ker(\phi)$ and $I \supseteq\ker(\phi)$, where $I$ is your desired ideal.)
The other useful hint here is that some of the easiest isomorphisms defined on $R[x]$ are the "point evaluations", i.e. substitute a fixed value for $x$ in each polynomial.
To illustrate this let's work out a very simple example showing that $R[x]/\langle x \rangle \simeq R$ for any ring $R$. Define $\phi : R[x] \rightarrow R$ by $ \phi: p(x) \mapsto p(0)$. It's a homomorphism because of the way the ring operations are defined in $R[x]$. It is onto because for any $a \in R$ we can define $p(x) = x + a \in R[x]$ and $\phi(p) = p(0) = a$. If $p(x) = x$ then $\phi(p) = 0$ so $\langle x \rangle \subseteq \ker(\phi)$. To prove the other inclusion, assume $q(x) \in \ker(\phi)$, so $q(0) = 0$, i.e. the constant term of $q$ is zero, so we can factor an $x$ out, i.e. we can write $q$ as $x\cdot p(x)$ for some $p$, so $q \in \langle x \rangle$, and this shows $\ker(\phi) \subseteq \langle x \rangle$. QED.
Notice that when we say we want to take $R[x]$ and mod out by $\langle x \rangle$ we can think of this as wanting to "set x equal to zero", To test your understanding, if we were given $R[x]/\langle x-1 \rangle$ we would want to set $x-1$ equal to zero. Which point evaluation would let us do that?
Now in your first problem you want to set $x^2 + x$ equal to zero. It's not quite as simple as the $R[x]/\langle x\rangle \simeq R$ example, but you can think of this as setting $x(x+1)$ equal to zero, so try to combine "setting $x$ equal to zero" and "setting $x+1$ equal to zero" to get a complete answer.
For your second problem you should be able to pick the appropriate point evaluation to handle "setting $x-2$ to zero" as your first step. But then you still have to handle the $x^2 + 1$, or more accurately, whatever $x^2+1$ maps to after you've applied your first homomorphism. Go ahead and figure out what your point evaluation, $\phi$, should be, and then tell yourself "Well, I want $\phi(x^2+1)$ to be zero", and see if that suggests anything to you.
(I've written this as a bunch of hints to help you get started and guide your thinking, so there are still a bunch of details for you to figure out and nail down. Feel free to follow up with what you figure out and requests if you feel you need further hints.)
