Rings whose modules have maximal or minimal injectivity domains.
Let \(M\) be a unital right \(R\)-module and \(R\) an associative ring with identity. The domain of injectivity of \(M\) is denoted by \(\text{In}^{-1}(M)\), i.e., \(\text{In}^{-1}(M)=\{N\in\text{Mod-}R\mid M\) is \(N\)-injective\}. Of course this domain always includes the semisimple modules; a module has been called poor if \(\text{In}^{-1}(M)\) consists only of semisimple modules [\textit{A. N. Alahmadi, M. Alkan} and \textit{S. R. López-Permouth}, Glasg. Math. J. 52A, 7-17 (2010; Zbl 1228.16004)]. Two constructions of poor modules are given, thereby showing that every ring has a poor module. A module \(M\) is said to crumble if the socles of all factors of \(M\) split. In the paper referred to above, certain semisimple modules that are poor were found; here it is shown that a ring \(R\) has a semisimple poor right module if and only if every (cyclic) right \(R\)-module that crumbles is semisimple, if and only if the only locally Noetherian right \(V\)-modules are the semisimple ones. In the paper of Alahmadi, Alkan and López-Permouth [see reference above], a ring \(R\) was defined to have no (right) middle class if each right \(R\)-module is either injective or poor. Investigating this concept further, it is shown that a ring with no right middle class is the ring direct sum of a semisimple Artinian ring and a ring \(T\) which is either zero or one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a ring \(V\)-ring, and such that \(\text{soc}(T_T)\) is homogeneous and essential in \(T_T\) and \(T\) has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In the third case, \(T\) is either a QF-ring with \(J(T)^2=0\) or poor as a right module. The proof is done by means of a series of lemmas which are also of interest on their own. For example, any ring \(R\) with no right middle class is either right semiartinian or right Noetherian, and if \(R\) is also nonzero and has a singular right socle, then \(R\) is right Artinian. Examples show that the three cases of the theorem are indeed possible. Some partial answers are obtained to the question if the inverse of the theorem discussed above is true. For example, if \(R\) is a right Artinian right SI-ring with homogeneous right socle and a unique (nonsingular) local module of length 2 up to isomorphism, then \(R\) has no right middle class. If \(R\) is a (non-semisimple) QF-ring with homogeneous right socle and \(J(R)^2=0\), then \(R\) is shown to have no right middle class.
scientific article; zbMATH DE number 5931654
scientific article