AbstractThe structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2ⁿ distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2ⁿ indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W.