I'm having trouble with a proof I've found on the web. Could you please help me? It's Lemma 2.0.1 here (Ash's Theorem, Finite Nilpotent Semigroups, and One-Dimensional Tiling Semigroups, D. B. McAlister).
The theorem reads:
Lemma 2.0.1 Let S be a finite nil semigroup. Then S is nilpotent. The nilpotent index of S is the maximal length of a strictly decreasing chain of principal non-zero ideals of S.
A nil semigroup is a semigroup with zero in which every element is nilpotent. A nilpotent semigroup a semigroup with zero for which there exists n>0 such that any product of n+1 elements of S is 0.
I understand the first part of the proof, where it is proved that a finite nil semigroup S is nilpotent, and that the nilpotent index is at most the maximal length of a strictly decreasing chain of principal non-zero ideals of S. What I don't understand is when the author says
"On the other hand, there is a strictly decreasing chain of non-zero principal right ideals of S."
I have no idea what this follows from.