Let $R$ be a ring and $I \subseteq R$ an ideal. How can you prove, that $R/I$ is a ring?
What could be an easy example for such an $R/I$ ?
I came up with $\mathbb{Z}/2\mathbb{Z}$ which should look like this:
$2\mathbb{Z} + 0 = \{..., -6, -4, -2, 0, 2, 4, 6, ... \}$
$2\mathbb{Z} + 1 = \{..., -7, -5, -3, 1, 3, 5, 7, ... \}$
Is this correct? If yes, I still need to show that all axioms of a ring are satisfied, right? (I would have to come up with $2^3$ terms to show that for example $+$ is associative, correct?)
Writing $[a]$ for the coset $a+I$, once you show that addition $[a]+[b]:=[a+b]$ and multiplication $[a]\cdot[b]:=[ab]$ in $R/I$ are well-defined, most required properties follow immediately from the respective properties in $R$. For example, $$\begin{align}([a]+[b])+[c]&=[a+b]+[c]\\&=[(a+b)+c]\\&=[a+(b+c)]\\&=[a]+[b+c]\\&=[a]+([b]+[c]) \end{align}$$