Volume 37 • Number 3 • 2014   • On A Class of $\delta$-Supplemented Modules Burcu Ungor, Sait Halicioglu and Abdullah Harmanci Abstract. Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module. In this paper, we introduce a class of modules which is an analogous to that of $\delta$-supplemented modules and principally $\oplus$-supplemented modules. The module $M$ is called {\it principally $\oplus$-$\delta$-supplemented} if for any $m\in M$ there exists a direct summand $A$ of $M$ such that $M = mR + A$ and $mR\cap A$ is $\delta$-small in $A$. We prove that some results of principally $\oplus$-supplemented modules can be extended to principally $\oplus$-$\delta$-supplemented modules for this general setting. Several properties of these modules are given and it is shown that the class of principally $\oplus$-$\delta$-supplemented modules lies strictly between classes of principally $\oplus$-supplemented modules and principally $\delta$-supplemented modules. We investigate conditions which ensure that any factor modules, direct summands and direct sums of principally $\oplus$-$\delta$-supplemented modules are also principally $\oplus$-$\delta$-supplemented. We give a characterization of principally $\oplus$-$\delta$-supplemented modules over a semisimple ring and a new characterization of principally $\delta$-semiperfect rings is obtained by using principally $\oplus$-$\delta$-supplemented modules. 2010 Mathematics Subject Classification: 32H05, 30C45 Full text: PDF