Incoming links: Hilbert's tenth problem
Sometimes systems of Diophantine equations are considered.
Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.
The general problem is known to be undecidable: Hilbert's tenth problem.
The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.
Unlike over non-commutative rings, polynomials do look like proper polynomials over commutative ring.
In particular, Hilbert's tenth problem is about polynomials over the integers, which is a commutative ring, and therefore brings mindshare to this definition.