My question is about the "right" way to think of prime numbers/elements.
Looking at primes in $\mathbb Z$, there are two ways of characterizing them:
- $p$ is prime iff its only divisors are $\pm 1,\pm p$
- $p$ is prime iff $p\mid ab\implies p\mid a\;\vee\;p\mid b$
The first characterization leads to defining irreducible elements while the second leads to the notion of prime element, which then leads to prime ideals and all sorts of nice things. I am tempted to say that prime elements are the "right" generalization of prime numbers because they are central to algebraic geometry. In this sense, maybe the "right" way to look at prime numbers is the second one, while the first characterization is just a coincidence because $\mathbb Z$ is a UFD.
Inspired by Zhen Lin's answer to this question let us look at a commutative ring $R$ as a category with $a\rightarrow b\iff a\mid b$. Then we can rewrite the two approaches as follows:
- In terms of arrows into it: the only arrows into $p$ are from its associates
- In terms of arrows from it: $p\rightarrow ab\implies p\rightarrow a\;\vee\;p\rightarrow b$
In fact, primes and irreducibles coincide in gcd domains, and being a gcd domain is the same (I think) as asking the category $R$ to have products. Hence, for irreducibles to be prime we generally need $R$ to have some completeness properties. On the other hand, primes are always irreducible.
One immediate way in which prime elements are "better" is that prime factorizations are automatically unique. To see this note the quotient by an ideal generated by a prime is an integral domain. Then take the equality $\prod_ip_i=\prod_jq_j$ to each quotient by $ \left\langle p_i \right\rangle$ to deduce equality.
But I wonder, is there perhaps a more convincing reason apriori to pick the second approach instead of the first? Maybe some fancy (maybe even philosophical) reason to look at arrows out of $p$? For instance, I thought the second approach makes more sense because it might give more arrows than the first.
Non-rigorous "philosophical" justifications are welcome too. I just want to get a feel for how people look at primes.
Ring theory has both the concept "prime element" and the concept "irreducible element". Each of these concepts is useful (or it wouldn't have been given a name), and it does not make sense to worry about which of the concepts is "better" or "right" without qualifying it with for such-and-such purpose.
The fact that the concept of prime elements has taken over the word "prime" from the elementary concept of prime numbers is an indication that the people who were most influential in selecting the naming considered "prime elements" to be a more useful generalization of prime numbers for the particular purposes they were concerned with.
This doesn't mean that one concept is "better" than the other in general, though. If you have an application in which you need to reason about irreducible elements, it won't be useful for you to decide to reason about prime elements instead just because they're supposedly "better".