Absolutely Pure Modules and Locally Injective Modules

Goro Azumaya asked the following question: if every absolutely pure left module over a ring R is locally injective, is R left Noetherian? The technique the authors’ make use of is to study the notions of absolutely pure module and locally injective module over almost maximal valuation domains. Over these rings absolutely pure modules turn out to be divisible modules, and locally injective modules turn out to be h-divisible modules. It is then easy to construct an example of an almost maximal valuation domain which is not Noetherian and over which there exists a divisible not h-divisible module. This example provides a negative answer to Azumaya’s question. Some of the results they make use of to answer Azumaya’s question have already appeared in the mathematical literature, but they provide some of the proofs in order to make the paper as self-contained as possible.