Almost injective modules lack a 'Baer-like' criterion

摘要

We provide a negative answer to a question raised by Jain and Alahmadi asking for a possible "Baer-like" criterion for almost injective modules. It is shown that there exists an R-module M which is almost injective relative to R but not almost injective. This is obtained by using our theorem: If every simple R-module is almost injective then every cocyclic module of length 2 is injective.