A variation of $\delta$-lifting and $\delta$-supplemented modules with respect to an equivalence relation

Document Type : Research Paper


Department of mathematics, Faculty of science and arts, Sinop University, Sinop, Turkey.



In this paper we introduce Goldie$^{\ast }$-$\delta $-supplemented modules as follows. A module $M$ is called Goldie$^{\ast }$-$\delta $-supplemented (briefly, G$_{\delta }^{\ast }$-supplemented) if there exists a $\delta $-supplement $T$ of $M$ for every submodule $A$ of $M$ such that $A\beta_{\delta }^{\ast }T$. We say that a module $M$ is called Goldie$^{\ast }$-$\delta $-lifting (briefly, G$_{\delta }^{\ast }$-lifting) if there exists a direct summand $D$ of $M$ for every submodule $A$ of $M$ such that $A\beta_{\delta }^{\ast }D$. Note that the last concept given in [4] as a $\delta $-$H$-supplemented module. We present fundamental properties of these modules. We indicate that these modules lie between $\delta $-lifting and $\delta $-supplemented modules. Also we prove that our modules coincide with some variations of $\delta $-supplemented modules for $\delta $-semiperfect modules.


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