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A note on tightness. (English) Zbl 0923.16003
For \(R\)-modules \(M\) and \(N\), \(M\) is called weakly \(N\)-injective if for every homomorphism \(\phi\colon N\to E(M)\), \(\phi(N)\subset X\simeq M\) where \(X\) is a submodule of the injective hull \(E(M)\) of \(M\). \(M\) is called \(N\)-tight if every quotient of \(N\) embeddable in \(E(M)\) is embeddable in \(M\). Every weakly \(N\)-injective module is \(N\)-tight. Generalizing earlier results of S. K. Jain and S. R. López-Permouth [J. Algebra 128, No. 1, 257-269 (1990; Zbl 0698.16012)] and S. K. Jain, S. R. López-Permouth and S. Singh [Glasg. J. Math. 34, No. 1, 75-81 (1992; Zbl 0747.16004)], the author proves that every cyclic right \(R\)-module is essentially embeddable in a projective module if \(R\) is right Artinian and every indecomposable projective right module is uniform and \(R\)-tight.

MSC:
16D50 Injective modules, self-injective associative rings
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