First of all, you need to know and fully understand what are the two operations defined on a particular
set that aspires to be a ring...and then you check all the axioms.
Certainy for the 3 example, say, there are several axioms that are automatic from a bigger set yours is contained in, say
commutativity (in this case of both addition and multiplication), associativity (since these two are true in ), etc., but
ANY non-obvious axiom must be fully checked.