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Erschienen in:
2016 | OriginalPaper | Buchkapitel
11 The Rhodes Radical and Triangularizability
verfasst von : Benjamin Steinberg
Erschienen in: Representation Theory of Finite Monoids
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Abstract
In this chapter we provide a correspondence between nilpotent bi-ideals and a certain class of congruences on a finite monoid. We characterize the largest nilpotent bi-ideal, which is called the Rhodes radical
because it was first described by Rhodes [Rho69b] in the case of an algebraically closed field of characteristic zero. For simplicity, we only give complete details in characteristic zero. As an application, we characterize those monoids with a faithful representation by upper triangular matrices over an algebraically closed field \(\mathbb{k}\) (the general case will be left to the reader in the exercises). These are precisely the monoids with a basic algebra over \(\mathbb{k}\) and so it also characterizes these monoids. Our treatment of these topics is based, for the most part, on that of Almeida, Margolis, the author, and Volkov [AMSV09].