An integral domain is called a Dedekind domain if it's not a field and every nonzero proper ideal admits a unique factorization into prime ideals. This is the most concrete way to say what a Dedekind domain is. But how do you check if a ring is a Dedekind domain? Emmy Noether found three conditions: if a domain is Noetherian, integrally closed, and one-dimensional then it's a Dedekind domain. Moreover the converse holds, so you can't make the number of hypotheses smaller in a non-artificial way. (In some references you will find those three conditions used as a definition of Dedekind domains.)