Pointwise Projective Modules and Some Related Modules

Let $\mathcal{R}$ be a commutative ring with 1, and Let M be a (left) R-module. M is said to be pointwise projective if for each epimorphism ${\alpha}:\mathcal{A}{\rightarrow}\mathcal{B}$, where A and $\mathcal{B}$ are any $\mathcal{R}$-modules, and for each homomorphism ${\beta}:\mathcal{M}{\rightarrow}\mathcal{B}$, then for every $m{\in}\mathcal{M}$, there exists a homomorphism ${\varphi}:\mathcal{M}{\rightarrow}\mathcal{A}$, which may depend on m, such that ${\alpha}{\circ}{\varphi}(m)={\beta}(m)$. Our mean concern in this paper is to study the relations between pointwise projectivemodules with cancellation modules and its geeralization.