It's only ambiguous insofar as one refuses to check the author's intended definition.
Here's another way to put it: with minimal to moderate effort, you can tell exactly what a term or symbol means in a given piece of mathematical writing. Whether this is by reading the definition or deducing the meaning from the form of the theorems (one can decide whether 'ring' includes '1' by the statements of theorems), it can be done.
This is why we aren't running around with contradictory and inconsistent mathematics: because working mathematicians know that they have to check what another mathematician means before they use their result.