Here is something I've wondered about since I was an undergraduate. Let $R$ be a ring (commutative, let's say, although the generalization to noncommutative rings is obvious). Ideals of $R$ can be ...
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field. What is a similar standard axiomatic definition of the integer numbers? A commutative ordered ...