I agree on all counts. But, the other answers that say that p is prime iff p is not a unit, and p | ab implies p | a or p | b, or, alternatively and equivalen

I agree on all counts. But, the other answers that say that p is prime iff p is not a unit, and p | ab implies p | a or p | b, or, alternatively and equivalently, p is prime iff p generates a prime ideal, are really the essential qualities of what makes something prime. And, even at that, most of the time, one doesn't care very much about whether individual elements of a ring are prime, but rather what are the prime ideals. The set of all prime ideals of a ring is called the spectrum of the ring, and can be used to impose a topology on it called the Zariski topology. This is useful for turning algebraic questions about a ring into topological questions that can be answered with topological methods. https://en.wikipedia.org/wiki/Prime_ideal https://en.wikipedia.org/wiki/Prime_ideal https://en.wikipedia.org/wiki/Zariski_topology#Spectrum_of_a_ring https://en.wikipedia.org/wiki/Zariski_topology#Spectrum_of_a...