G– $$\delta$$ δ –M Modules and Torsion Theory Cogenerated by Such Modules

Abstract

Let N be a module in category \(\sigma [M]\). N is called generalized \(\delta\)–M-small (briefly G–\(\delta\)–M) if, \(N \subseteq \delta (L)\) for some \(L \in \sigma [M]\). In this paper we characterize G–\(\delta\)–M modules and get some suitable results related to this kind of modules. We will show that a module \(N \in \sigma [M]\) is G–\(\delta\)–M if and only if \(N \subseteq \delta (\hat{N})\), where \(\hat{N}\) is the M-injective envelope of N in \(\sigma [M]\). Also we prove that if there is no non-zero G–\(\delta\)–M module, then M is cosemisimple and by giving an example we show that the converse need not be true. The relation between G–\(\delta\)–M modules and some other classes of modules would be investigated in this paper. The torsion theory cogenerated by this class of modules will be introduced and studied in this paper. For a module \(N \in \sigma [M]\) we show that \(N = \mathrm{{Re}}_{GD[M]}(N)\) iff for every nonzero homomorphism \(f : N \longrightarrow K\) in \(\sigma [M],\) \(\mathrm{{Im}}(f) \not \subseteq \delta (K)\) iff \(\Delta _{\delta }(N,A) = 0,\) for all \(A \in \sigma [M]\).