The last post I made had a little terminology mix-up, but I liked reading the responses nevertheless.
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Rings you can’t necessarily divide by nonzero elements; fields you can.
One ring that’s “not numbers” is the ring of polynomials (with real coefficients) in one variable X. You can add, subtract, and multiply them — we teach middle-schoolers to do it — but you can’t divide polynomials and get a polynomial result. Thus, this isn’t a field.
However, what if we consider two polynomials to be “the same” if they take the same value at X=0. That is, if they have the same constant term. This is a quotient ring of the ring of polynomials, and it consists of equivalence classes of polynomials under this relation. Of course, it just happens to be isomorphic to the real numbers, so maybe you don’t like that example.
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