What type of thing is N? What you wrote doesn't seem to make sense if N is just a constant, but I don't see how it makes sense if N is a function? Or maybe you didn't mean multiplication?
E.g. N = 3, x = 2, y = 3
N(xy) = 3(2 . 3) = 18
N(x)N(y) - N = 3(2) . 3(3) - 3 = 51, which is not 18
That's not a subtraction sign, it's a hyphen. N is the norm of x, denoted by |x|. Technically the norm is a scalar-valued function applied to a vector, hence the functional notation N(xy) = N(x) • N(y).
So to be explicit, they're saying |xy| = |x| • |y| implies you cannot have a 0 divisor.
N here is a function, not a constant. I was confused at first, but I think the "-N" in that post wasn't intended to mean "minus N" but rather ", where N is the function called...".
The "norm" of a number is more or less its absolute value: its size, it's magnitude. So "1" has norm 1, but "-1" also has norm 1, as does "i", and "-5" or "5i" or "4-3i" all have norm 5. So these four mathematical structure (R,C,Q, and O) all have the property that the norm of a product is equal to the product of the norms. Things get really obnoxious (or at least really unfamiliar) if you don't have that property.
Edit: Sorry for the repetition! I've got to remember to reload these pages before replying.
E.g. N = 3, x = 2, y = 3
N(xy) = 3(2 . 3) = 18
N(x)N(y) - N = 3(2) . 3(3) - 3 = 51, which is not 18