Note on almost \(M\)-injectives (Q913912)

The author defines the concept of ``almost \(M\)-injectives'' as a dual to the ``almost \(M\)-projectives'' recently defined by Harada and Tozaki. Almost \(M\)-injectivity generalizes the well-known \(M\)-injectivity defined by Azumaya and the purpose of the paper is to generalize Azumaya's theorem on \(M\)-injectives stating that \(N\) is \(M_1\)- and \(M_2\)-injective if and only if \(N\) is \(M_1\oplus M_2\)-injective (where \(N\), \(M_1\), \(M_2\) are \(R\)-modules). This property is shown not to be true if ``\(M\)-injective'' is just replaced by ``almost \(M\)-injective'' but the proof of a generalization with additional conditions constitutes the greatest part of the paper.