It kind of depends on how you define prime numbers. Usually, one says an element p of a general ring is prime if p|ab implies p|a or p|b. This property does no

It kind of depends on how you define prime numbers. Usually, one says an element p of a general ring is prime if p|ab implies p|a or p|b. This property does not necessarily hold for the "smallest" divisor of a given element of your ring, as used in proof 1. On the other hand, you might be interested in irreducible elements, i.e. those p for which a|p implies a = 1 or p, up to a unit (i.e. a divisor of 1). In fact, you can write any element of your ring as a product of irreducible elements, but not uniquely, not even uniquely up to units. In other words, you can have irreducible elements a, b, c, d, such that ab = cd, but none of them divides any of the others.