I am currently reading the Artin's Algebra, in Chapter 11, section 4, I met some something hard to understand.
1) Adding relations: In this subsection, the book states that "let a and b be elements of a Ring $R$ and let $\bar{R} = R/(a)$ be the result of killing a in $R$. Let $\bar{b}$ be the residue of b in $\bar{R}$. The Correspondence Theorem tells us that the principal ideal $(\bar{b})$ of $\bar{R}$ corresponds to the ideal $(a,b)$ of $R$". My question is what does "Let $\bar{b}$ be the residue of b in $\bar{R}$" mean? I don't know how to use correspondence theorem to show this fact.
2)In the following example, I don't know why $\mathbb{Z}[x]/(x-2,x^2+1)$ is isomorphic to $\mathbb{F}_5$. But I know that $\mathbb{Z}[x]/(x-2)$ is isomorphic to $\mathbb{Z}$.
In terms of cosets, $\bar b=b+(a)$. In terms of equivalence classes, $\bar b = [b]$, i.e. the equivalence class that $b$ represents.
For a commutative ring, $R$, an ideal $I$ gives is equivalent to a congruence, $\sim$, on $R$ such that $x \sim y \iff x-y\in I$. The elements of $R/I$ are then elements of $R/{\sim}$, i.e. equivalence classes of $R$ with respect to $\sim$. The upshot $I$ is the equivalence class of $0$, i.e. $I=[0]$ in $R/I$. Since, in your example, $x - 2 \in I$ and $x^2+1\in I$ by definition, we have $[x-2]=[x]-[2]=[0]$ and $[x^2+1]=[x]^2+[1]=[0]$. If we combine these equations, we get $[x]=[2]$ and thus $[5]=[0]$. The result is essentially equivalent to $\mathbb Z/(5)$ which is $\mathbb F_5$.
(I think $\bar R$ was supposed to be $\bar R = R/(a)$, not $\bar R/(a)$.)