Definition of primality that is abstract-algebra-friendly

Question

I have tried to give my own definition of "prime element" that fits nicely with how I see abstract algebra. Here it is:

Given an algebraic structure of this form (including any other axioms):

$$ \mathcal{A} = (S, \cdot, 0_S)$$

where $S$ is a set, $\cdot$ is a binary operation, and $0_S$ is the identity for the operation, s.t.

$$\forall p \in S, 0_S \cdot p = p \cdot 0_s = p$$

we define a prime element this way.

Given $p \in \mathcal{A}$ and $F_p \subset S \times S$ s.t. for $\forall (f1, f2) \in F_p, f1 \cdot f2 = p$, then we call $p$ a prime element iff $F_p = \{(0_p, p) \}$.

Now, we can do proofs like this: Do the integers contain a prime element? No.

proof:

$$\mathcal{A} = (\mathbb{Z}, 0, +)$$

We have the added axiom of inverses: $\forall a \in \mathcal{A}, \exists a^{-1}$ s.t. $ a + a^{-1} = 0$

We have the added axiom of closure: $\forall a,b \in \mathcal{A}, \exists c \in \mathcal{A}$ s.t. $a + b = c$.

By closure, $\forall p,a \in \mathcal{A} \exists b \in \mathcal{A} $ s.t. $ p + a =b$

Then $p = b + a^{-1}$ and by inverses $a^{-1} \in \mathcal{A}$

Then $|F_p| = \mathbb{Z}$

and so $ F_p \neq \{(0, p) \} \forall p $

We want to now do a proof that natural numbers with multiplication contains primes.

$\mathcal{A} = (\mathbb{N}, \times, 1)$

This has all the base axioms we need. We want to use the property that this is actually a commutitave monoid on countably many generators. The proof is trivial because the generators are the primes and we just say that generators must have $F_p = \{ (1, p) \}$.

My question is, what is wrong with my definition of prime element? If it is acceptable, can we use this as a basis to describe which algebraic structures will have "prime elements" according the definition?

Answer 1

A few technical nits:

• It's misleading to use the symbol $0$ for the identity element of an operation called $\cdot$. It would be better to call the identity $1$ or $e$. The $0$ symbol should be reserved for an absorbing element.
• Putting a subscript on the identity element $1$ but not on the binary operator $\cdot$ is somewhat inconsistent. It's a sign that you should consider abandoning subscripts entirely, trusting the reader to use the context to understand what the identity element is for. Doing so will save you from various mistakes above: writing $0_s$ versus $0_S$ versus $0_p$ which are supposed to be the same.
• The core definition of primality contains a mistake: the cartesian product $S\times S$ contains ordered pairs, not unordered pairs, so the elements $(1,p)$ and $(p,1)$ are different. You use the singleton set $\{(1,p)\}$ where you probably want the set $\{(1,p), (p,1)\}$, which has two elements if $p\neq 1$.
• The order of quantification in the usage of $F_p$ is also a bit muddled. Instead of saying:

Given $p \in \mathcal{A}$ and $F_p \subset S \times S$ s.t. for $\forall (f1, f2) \in F_p, f1 \cdot f2 = p$, then we call $p$ a prime element iff $F_p = \{(0_p, p) \}$.

You meant something more like:

Given $p \in \mathcal{A}$, we define $F_p \subset S \times S$ to be the set of all $(f1, f2) \in F_p, f1 \cdot f2 = p$. Now, we call $p$ a prime element iff $F_p = \{(0_p, p) \}$.

In your version, $F_p$ is not uniquely determined by $p$. In fact, for any $p$, we can set $F_p=\{(0_p, p) \}$, and then $p$ and $F_p$ satisfy the condition by fiat, so we should call every $p$ a prime.

• When you describe $\mathcal A$, regarding "We have the added axiom of closure", it's not correct to think of that as an added axiom. The axiom of closure is implicit in the definition of an algebraic structure.

Now for a more substantive discussion... The main things missing from the definition are motivation and comparison with the literature. There is already the notion of a "prime" element of a ring, and of an "irreducible" element in various more general settings. Your new definition is not quite the same as the ones I linked to. You'll want to argue: when does this new definition pick out a more useful property than the existing definitions? And why should it be called "prime", rather than "irreducible" or a totally novel word?