A General Approach to the Fitting Lemma

Abstract

Results are formulated about the image and the kernel of the kth iterate fk of a function f : A → A. In this way, an extremely general version of Fitting's classical lemma is obtained. Two applications are presented: the first is a characterization of strongly π‐regular rings, while the second is a “lattice theoretical Fitting lemma”.

Linear algebra in lattices and nilpotent endomorphisms of semisimple modules

“…Theorem 3.7 is the generalization of the well-known Fitting lemma (see [1, p. 138]). The lattice theoretical Fitting lemma in [3] is an analogue and not a generalization of the classical result.…”

“…The purpose of this paper is to show, how the above classical result is capable of broad generalization in the context of lattices. The present work is of the same flavour as the job we carried out in [2]. We consider a complete lattice L and a nilpotent ∨-homomorphism λ : L −→ L. The Jordan normal base of L with respect to λ is defined in a natural way.…”

“…Then we formulate the Fitting lemma for λ. We note that a completely different lattice theoretical Fitting lemma for lattice endomorphisms can be found in [3]. If λ is nilpotent, then an appropriate definition of the Jordan normal base of L with respect to λ can be given.…”