Factorization in the Sense of Primes

Question

Are there types prime factorizations in the sense of primes other than the Fundamental Theorem of Arithmetic (and its generalization to euclidean/principle rings), primary decomposition and Dedekind domains?

Answer 1

Absolutely, but then it becomes essential to distinguish between irreducible and primes, as they are not always the same.

For a non-unit number to be irreducible, it's enough that it's not the product of two other non-unit numbers. But for it to be prime, it must be divide at least one factor of any product that it's a divisor of.

For example, in $\mathbb Z[\sqrt{10}]$, we see that 2, 3, $2 - \sqrt{10}$ and $2 + \sqrt{10}$ are all irreducible but not prime, since $2 \times 3 = (-1)(2 - \sqrt{10})(2 + \sqrt{10}) = 6$, yet in this ring $2 \nmid (2 \pm \sqrt{10})$, $3 \nmid (2 \pm \sqrt{10})$.

However, 7 is prime in this domain. You can find nonzero multiples of 7 with more than one factorization, but in every case you will find 7 irreducible in those factorizations.

Not quite sure if that's what you're looking for. Maybe this: a domain "smaller" than $\mathbb Z$, which consists solely of positive integers of the form $4k + 1$. If we start with $\mathbb Z^+$, we remove integers of the forms $4k$, $4k + 2$ and $4k + 3$ so that we are left with 1, 5, 9, 13, 17, 21, 25, 29, etc.

Clearly 1089 is composite in this domain. Its prime factorization is $3^2 \times 11^2$. Oops, that's wrong, since neither 3 nor 11 is of the form $4k + 1$. Instead it's factorizations are $9 \times 121 = 33^2$. Therefore 9, 33 and 121 are all irreducible but not prime.

For a number to be prime and irreducible in this domain, it has to be a prime of the form $4k + 1$ in $\mathbb Z$. So 5 is prime and irreducible in this domain, sometimes called $\mathcal S$. We see, for example, that 2205 has more than one distinct factorization in this domain, but the blame for that goes to 441, not 5.

And here is one where things get really trippy: the finite $\mathbb Z_{10}$, consisting entirely of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Addition and multiplication are adjusted to keep arithmetic within $\mathbb Z_{10}$, e.g., $5 + 5 = 0$. Then 5 is prime but not irreducible, as $5 = 5^2 = 5^3 = 5^4 = \ldots$