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Poor modules: the opposite of injectivity. (English) Zbl 1228.16004

For a right \(R\)-module \(M\), its domain of injectivity, denoted by \(In^{-1}(M)\) is defined as \(In^{-1}(M)=\{N\in\text{Mod-}R:M\) is \(N\)-injective}. It is not difficult to see that if \(S\) is a semisimple right \(R\)-module, then \(S\in In^{-1}(M)\) for any right \(R\)-module \(M\). Clearly, \(M\) is injective if \(In^{-1}(M)=\text{Mod-}R\).
The paper under review studies those modules whose domain of injectivity is as small as possible. The authors call a right \(R\)-module \(M\) a poor module if \(In^{-1}(M)\) is the class of semisimple right \(R\)-modules. A ring \(R\) is said to have no middle class if every right \(R\)-module is either poor or injective. If every right module over a ring \(R\) is injective, or if every module is poor, then \(R\) is semisimple Artinian. However, rings with no middle class are not necessarily semisimple Artinian. For example, if \(R\) is a right PCI domain then \(R\) has no middle class and \(R_R\) is a poor module.

MSC:

16D50 Injective modules, self-injective associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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