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On generalized extending modules. (English) Zbl 1156.16005
Summary: H. Hanada, J. Kado, and K. Oshiro have introduced, in a diagram of modules and homomorphisms, the concept of generalized \(M\)-injective modules. S. Mohamed, and B. Müller have given a different characterization, based on an exchange property, of the generalized \(M\)-injective modules. Here we introduce the concept of \(M\)-jective modules, which is a generalization of Mohamed and Müller’s concept for the generalized \(M\)-injectivity. The concept of \(M\)-jective modules is used here to solve the problem of finding a necessary and sufficient condition for a direct sum of extending modules to be extending. In fact, we show that relative jectivity is necessary and sufficient for a direct sum of two extending modules to be extending. We also introduce the concept of generalized extending modules, and give some properties of such modules is analogy with the known properties for extending modules.
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D50 Injective modules, self-injective associative rings
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