  In this study, all rings are commutative with non-zero identity and all
modules are considered to be unital. Let $M$ be a left $R$-module. A proper
submodule $N$ of $M$ is called an $S$-$weakly$ $prime$ submodule if $0_{M}\neq
f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in
S=End(M)$ and $m\in M$. Some results concerning $S$-prime and $S$-weakly prime
submodules are obtained. Then we study $S$-prime and $S$-weakly prime
submodules of multiplication modules. Also for $R$-modules $M_{1}$ and $M_{2},$
we examine $S$-prime and $S$-weakly prime submodules of $M=M_{1}\times M_{2},$
where $S=S_{1}\times S_{2},$ $S_{1}=End(M_{1})$ and $S_{2}=End(M_{2})$.
