I'm taking a course which introduces modules this semester. In the notes, it gives the definition of a module (or left module) over a ring (where we generally always assume rings have multiplicative identities) as an abelian group with an extra map (the action of the ring) with the usual four properties (this is the definition you'd find on Wikipedia say).
After giving the definition, it makes this remark:
The axiom $1m = m$ ($1$ being the multiplicative identity of the ring and $m$ being any element of the module) means that the identity element $1$ acts on the module as the identity map. If this were not true, we would be in a state of notational dissonance, which we would have to resolve by renaming the identity element of the ring, say as $e$.
That I'm okay with, but then it continues to say:
But then for all practical purposes it would make no difference to throw a new element called $1$ into our ring, which satisfied $1r = r1 = r$ for all $r$ in the ring, and $1m = m$ for all $m$ in the module. The same would apply if we had allowed the ring not to have a multiplicative identity previously. This is why we are not really losing any generality by assuming that $R$ has an element $1$ and that it acts on the module as the identity.
Here, I don't understand why you could just freely throw a new element in the ring? Would we not have to worry about violating the ring axioms? And even if we ignored them, how do we know the four properties for the action map would still remain intact?