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Characterizations of lifting modules in terms of cojective modules and the class of \(\mathcal B(M,X)\). (English) Zbl 1082.16005

Let \(M\) and \(X\) be two \(R\)-modules and let \({\mathcal B}(M,X)=\{A\leq M\mid\exists Y\leq X\), \(\exists f\in\operatorname{Hom}(M,X/Y)\), \(\text{Ker\,}f/A\ll M/A\}\). Then \(M\) is called \(X\)-lifting if, for every submodule \(A\) of \(M\) with \(A\in{\mathcal B}(M,X)\), there exists a direct summand \(B\) of \(M\) such that \(A/B\ll M/B\) (S. H. Mohamed and B. H. Müller [Preprint]). A module \(Q\) is said to be \({\mathcal B}(M,X)\)-cojective if, for any submodule \(A\) of \(M\) with \(A\in{\mathcal B}(M,X)\) and any homomorphism \(\varphi\colon Q\to M/A\), there exist decompositions \(Q=Q_1\oplus Q_2\), \(M=M_1\oplus M_2\) and homomorphisms \(\psi_1\colon Q_1\to M_1\), \(\psi_2\colon M_2\to Q_2\) such that \(\psi_2\) is onto, \(\pi\psi_1=\psi| Q_1\) and \(\psi\psi_2=\pi| M_2\), where \(\pi\colon M\to M/A\) is the natural epimorphism.
The paper first studies the properties of \({\mathcal B}(M,X)\)-cojective modules and the related concepts of small \({\mathcal B}(M,X)\)-cojectivity and pseudo \({\mathcal B}(M,X)\)-cojectivity. Several characterizations of these properties are found. Then \(X\)-lifting modules are studied. Let \(A\) and \(P\) be submodules of \(M\) with \(P\in{\mathcal B}(M,X)\). Then \(P\) is called an \(X\)-supplement of \(A\) if it is minimal with respect to the property \(A+P=M\). A module \(M\) is called \(X\)-amply supplemented if for any submodules \(A,B\) of \(M\) with \(A\in{\mathcal B}(M,X)\) and \(M=A+B\), there exists an \(X\)-supplement \(P\) of \(A\) such that \(P\leq B\). Various sufficient conditions and equivalences for an amply supplemented module \(M\) to be \(X\)-lifting are found. For example, if \(M=M_1\oplus M_2\) is an \(X\)-amply supplemented module, then \(M\) is shown to be \(X\)-lifting iff every \(X\)-supplement submodule \(K\) of \(M\) such that either \(M=K+M_1\), or \(M=K+M_2\), is a direct summand of \(M\).

MSC:

16D80 Other classes of modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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