[gus_massa]

At least in my university, in one of the first Algebra courses about integers, factorization and congruence the usual notation is

  3 * 3 ≡ 1 (mod 4)

but in the next year Algebra course about groups and crazy algebraic structures the notation in the group Z_4 is just

  3 * 3 = 1

and everyone understand that you are working in Z_4 (and the * and = symbols are "overloaded" (but no one call them "overloaded")).

[OscarCunningham]

I don't think "=" is being overloaded in this example. The symbols "1", "3" and "*" are, since they're working in Z_4 rather than Z, but equality is just equality.

[lvh]

Sure; I think we’re in violent agreement here, it’s absolutely the case that people write the simpler version when the meaning is clear from context. I’ve definitely done that a bunch.

[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.