A generalization of supplemented modules

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 of $\delta$-supplemented modules defined by Kosan. The module $M$ is called principally $\delta$-supplemented, for all $m\in M$ there exists a submodule $A$ of $M$ with $M = mR + A$ and $(mR)\cap A$ $\delta$-small in $A$. We prove that some results of $\delta$-supplemented modules can be extended to principally $\delta$-supplemented modules for this general settings. We supply some examples showing that there are principally $\delta$-supplemented modules but not $\delta$-supplemented. We also introduce principally $\delta$-semiperfect modules as a generalization of $\delta$-semiperfect modules and investigate their properties.