This is a fascinating section of Hungerford’s book. Here he starts to accumulate all of the ideas collected so far on rings and unite them. He starts to do so with an ideal.
Theorem 6.1
A nonempty subset I of a ring R is an ideal if and only if it has these properties

if , then ;

if and , then and
When I first read this the first hing I noticed is that an ideal has the same characteristics as a ring, that is a ring is closed under subtraction and multiplication. It’s incredible how linked these ideas are with the integers and the polynomials that they can be generalized to be valid for any ring. Why generalize it though? What rings have not been considered that now can be?