Look at the number 6. If you only consider the positive integers, then the prime factorization is 2 and 3. That’s it. A unique prime factorization.
Now look at all integers. Now the factorization could be 2 and 3 or -2 and -3. Now suppose you look at all numbers of the form (a + b*sqrt(-5)) where a and b are integers. You can prove that these numbers form a closed set. In fact, it forms the Z[sqrt(-5)] ring. Suddenly, (1-sqrt(-5)) and (1+sqrt(-5)) multiply to 6. Both of these number cannot be broken down any further, so they are prime. 6 no longer has a unique prime factorization. If you set b = 0, you’ll notice that 2 and 3 are also in this ring, so the original unique prime factorization is also here.
This is the beginning of prime ideals (due to people like Dedekind). If you have a ring, which has a bunch of numbers, you can look at subsets, usually infinite subsets of the ring. For example, suppose I have subset A and subset B. Now, although 6 does not have a unique prime factorization, suppose that A*B contains all of its unique prime factorizations, then we say that 6 has a unique prime ideal factorization. And now prime factorization has been “saved” or “re-established.”
