ac.commutative algebra - How to design or create or generate a bijective ring map?

In generality (this is tagged "commutative algebra", so let's talk commutative rings) I wonder if there is more than taking generators of each side and writing the images as polynomials in the generators of the other side. This is a candidate for a bijection of rings, but so far isn't a homomorphism (you need to check the relations hold). In other words express each ring as a quotient of a polynomial ring, set up homomorphisms from the polynomial rings, and then show the kernels are what they should be.