Do all rings have to be generated by ideals? Or can some rings come without ideals? Can some elements in rings be generated by ideals (in ways that other elements of rings are untouched by ideals?) If ALL of a ring's elements are generated by ideals, is there something special about the ring? (for example, the ring of Gaussian integers is completely generated by around 6 ideals IIRC). If an element in a ring is hit on by two different ideals, is there anything wrong with that?