  The authors of [Primitive orthogonal idempotents for R-trivial monoids,
Journal of Algebra] provide an algorithm for finding a complete system of
primitive orthogonal idempotents for CM, where M is any finite R-trivial
monoid. Their method relies on a technical result stating that R-trivial
monoids are equivalent to so-called weakly ordered monoids. We provide an
alternative algorithm, based only on the simple observation that an R-trivial
monoid may be realised by upper triangular matrices. This approach is inspired
by results in the field of coupled cell network dynamical systems, where
L-trivial monoids (the opposite notion) correspond to so-called feed-forward
networks. We also show that our algorithm only has to be performed for ZM,
after which a complete system of primitive orthogonal idempotents may be
obtained for RM, where R is any ring with a known complete system of primitive
orthogonal idempotents. In particular, the algorithm works if R is any field.
