Honestly saying you can't have a number 1/0 because it breaks the ring axioms seems exactly analogous to saying 0 can't exist because it breaks the group axioms for multiplication on the non-zero reals. Is ring multiplication "not really multiplication" because it doesn't satisfy group axioms? That doesn't seem consistent with normal usage to me, but you could imagine a pedantic student coming out of their first group theory course and trying to make that argument.
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Honestly saying you can't have a number 1/0 because it breaks the ring axioms seems exactly analogous to saying 0 can't exist because it breaks the group axioms for multiplication on the non-zero reals. Is ring multiplication "not really multiplication" because it doesn't satisfy group axioms? That doesn't seem consistent with normal usage to me, but you could imagine a pedantic student coming out of their first group theory course and trying to make that argument.