Hi /calc/,

I'm puzzling over the notion of a size function in my ring theory class. Since my prof doesn't seem to be holding office hours yet I thought I'd bring this here and hope others may benefit as well.

For R an integral domain, a size function N: R --> {non-negative integers} was defined to have the following properties:
(i) N(0) < N(r) for all r non zero
(ii) There exists an m such that N(r) = m iff r is a unit. Moreover N(r) > m for all r not a unit and non-zero.
(iii) For a, b in R non-zero, N(a) < N(ab) and N(a) = N(ab) iff b is a unit.

I'm wandering if my following thoughts are correct or if I'm funamentally mistaken about something here. If I suppose r in R has a non trivial factorization, r = sq, then by (ii) it could be that N(r) = N(sq) = m+1. But s and q are non-unit and non-associates of r, either they're irreducible or have further non-trivial factorizations themselves. Would be it more appropriate to say the "smallest" non trivial factorization can have size 2(m+1) = 2m+2 for this reason? Additionally, what if r isn't a unit, but is irreducible? Does it necessarily have size < m?. Or can the second part of property (iii) be weakened to say something like "N(a) = N(bc) for b and associate of a and c a unit"?

I'm trying to figure out how I might use N to show that r with N(r) = m+1 has a non trivial factorization and ran into these questions. I feel like there isn't enough information here to go off of. Or I'm just too dumb to see the obvious. IDK

Thank you