I was glad that the article got into the definitions of irreducible, prime, and unit in more general algebraic structures! This question is all about these thre

I was glad that the article got into the definitions of irreducible, prime, and unit in more general algebraic structures! This question is all about these three concepts. A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1. So these are units in the integers. Units are really important, but they don't get you anywhere on their own. We can think of non-units as building blocks. So then the question comes up which building blocks can be broken down into smaller blocks, like "times 6" can be divided into "times 2, then times 3". Using a unit doesn't count, because you're still equally far from the starting point (so 6 = (-6)(-1) is not really breaking down 6). Blocks that can't be broken down any farther are irreducible (like 7). But in common parlance, that's what we think of as the definition of a prime number. It doesn't include units like 1, because they're already broken down. More generally, being prime isn't about being broken down, it's about building up. Suppose I tell you that n is divisible by 2, and n can be broken down into two pieces, n = ab. Then you know that at least one of a or b is divisible by two. So two is prime because anything that is built from two, when broken down into pieces, has a piece that can be built from two. (Example: 6 divides 12, but you can break 12 up into (3)(4), so 6 isn't prime.) This definition of prime also doesn't include units like 1 because they're not building blocks in the first place. Everything's already divisible by units. Prime integers have both the prime property and the irreducible property. Either way, I think it makes sense that 1 isn't included -- being prime and irreducible are properties of building blocks, and 1 is a unit.