On generalized Nakayama–Azumaya’s lemma

Abstract

In the recent publication, generalized Nakayama–Azumaya’s lemma, i.e. Azumaya’s lemma (or Nakayama’s lemma) extended to a module which is isomorphic to a direct summand of a direct sum of finitely generated modules, is conjectured to hold. A proof of this conjecture is the main objective of the present paper. It is achieved by proving the nonexistence of non-zero weak Nakayama–Azumaya special modules, and showing that the canonical map is a non-split epimorphism if Mi is finitely generated and for each As an application of generalized Nakayama–Azumaya’s lemma, we show the existence of maximal submodules for some classes of modules. This implies, in particular, Bass’ result concerning the class of projective modules.