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 $f:{\oplus }_{i\in \mathbb{N}}{M}_i\to M=\underset{i\in \mathbb{N}}{\cup }{M}_i$ is a non-split epimorphism if M_i is finitely generated and $M_i\subset M_{i+1}J(R)$ for each $i\in \mathbb{N}.$ 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.

Keywords:

• Azumaya’s lemma
• existence of maximal submodule
• Nakayama–Azumaya special (NAS) module

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 08A05; 16S90;
• Secondary: 16D10; 16N20; 16U99

Acknowledgments

The author expresses great thanks to Professor Kunio Yamagata for his helpful advice. The author thanks also to Professor Vlastimil Dlab for a number of corrections and improvements to the final text.