What is Quotient ring Z/nZ, revisited?

For an ideal $I$ of $A$, the relation $x \sim y \iff x - y \in I$ is an equivalence compatible with $+$ and $\times$; the quotient set $A/I$ inherits a ring structure — the quotient ring — making $\pi \colon A \to A/I$ a morphism with kernel $I$. For $A = \Z$, $I = n\Z$ this is the $\Z/n\Z$ of the Year 1 volume, now with its universal property: any morphism killing $I$ factors through $A/I$.