On nice modules (Q2826318)

An associative ring with \(1\) is called a \textit{clean ring} if each of its elements is the sum of a unit and an idempotent. These rings were introduced by Nicholson as examples of exchange rings. Generalizing this definition, a module is called \textit{clean} if its endomorphism ring is a clean ring. It is known that all continuous modules are clean, hence so are injective modules.NEWLINENEWLINEIn the paper under review, the authors introduce the notion of a \textit{nice module}, by which they mean every submodule is clean. There are three main results in the paper. First, observe that if a module-theoretic property \(P\) entails the clean condition and passes to all submodules, then modules with \(P\) are nice. The authors describe a few such properties, including ``finite length'' and ``semisimplicity.'' Second, they show that these specific properties are not necessary for nice modules. For instance, the Prüfer \(p\)-groups are nice but not semisimple nor of finite length. Third, they point to examples existing in the literature which show that the artinian, noetherian, and injective properties (separately) do not imply that a module is nice. (The first two properties do not even imply the clean, or even exchange, property.)