I am a bit confused, as my class is currently on Section 9.2 of Dummit and Foote, and what elements of $R/I$ are of the form of, where $R$ is a polynomial ring.
For example, I am reading that $\mathbb{Z}[x,y]/(x^2,y^2,2)$ are of the form $a + bx + cy + dxy$, where $a,b,c,d \in \mathbb{Z}_2$. How is this deduced?
I suppose there is a definition $R/I = \{ r + I, r \in R \}$, but how does this form $a + bx + cy + dxy$, and can anyone give some other examples of how we can write elements of $R/I$ with a polynomial ring?
How should I be thinking about this more deeply?
You can think of quotienting by an ideal as a way of making the elements in the ideal become $0$. A simple example is $\mathbb{Z}/6\mathbb{Z}$ which is just the ring with elements $\{0,1,2,3,4,5\}$. We've made $6 = 0$ which in turn meant that $7 = 1, 8 = 2, ...$. In your example we are making $x^2, y^2$ and $2$ equal $0$. Having $x^2$ be $0$ means that any term of the form $ ax^ny^m = 0$ if $n \ge 2$. We have a similar situation for $y^2$. This means that your polynomial can only have the form $ a + bx + cy + dxy $ since the higher order terms will be $0$. The reason why the coefficients are in $Z_2$ is because we have quotiented by $2$ which means $2 = 0$. Hence all even numbers are actually $0$ since they can be written as $2n = 2\cdot 0 = 0$. All odd numbers are just equal to $1$ since they can be written as $2n + 1 = 2 \cdot 0 + 1 = 1$.