[mclehman]

Piling on with a bit more pedantry, my experience is a bit different.

In my current ring theory course, we have indeed written things like 3 * 3 = 1 when working in |F_5 (not sure that notation is going to work as well as I hope, looks alright in the app I use), but it's not the equality symbol is overloaded, but the numbers themselves. Rather than using = to mean numeric equality and equality w.r.t. equivalence classes, we just use the numbers themselves as shorthand for their equivalence classes.

[laingc]

That seems odd to me. I don't think I've read any ring/algebra/module theory text that doesn't explicitly denote equivalence classes with, for example, square brackets.

[heinrich5991]

There's a canonical ring homomorphism from the integers into any commutative ring with 1. When such a homomorphism is unique, you often omit it, hence mathematicians sometimes just write numbers without equivalence class brackets.

This is not limited to rings of the form Z/nZ.