Ideals Give Us Factor Rings

When you assume the first bullet above, you get to assume that the operations are binary operation. However, when you assume the second bullet, you must prove that the operations are binary operations. If we let $x,y\in R/A$, then there exists $s,t\in R$ such that $x=s+A$ and $y=t+A$. However, the choice of $s$ and $t$ is not unique. Suppose also that $x=s'+A$ and $y=t'+A$. The operation given for computing products then says that $xy=(s+A)(t+A) = st+A$, but it also says that $xy=(s'+A)(t'+A) = s't'+A$. However, if $st+A\neq s't'+A$, then we have a problem, as then $xy$ is not well-defined. This is what you must show. You must prove that if $A$ is an ideal, then you are guaranteed that $(s'+A)(t'+A) = s't'+A$.