Why is subtraction not part of the algebra? It’s certainly familiar to every high school math student. This omission allows the counterexample, so the reveal is a bit of a disappointment IMHO.
(Edit: positive integers aren’t exactly a semiring because 0 is excluded, although some authors do define a semiring without the requirement of an additive identity element.)
If we allow negatives but disallow exponentiation, that's decidable. If we allow negatives and partial exponentiation that stays within the integers, that's undecidable again.
Similarly, I think a * b and a / b can be replaced with the same trick, but then I realized it may not work on non-abelian, or where multiplicative inverse is not available...