Simple $R$-module equivalent statement

Question

I am trying to show that the two statements are logically equivalent.
a) $X$ is a simple $R$-module.
b) For any $x , y in X$, $x,y neq 0$ , there exists $r in R$ such that $rx = y$.
In other words,  I am trying to show that $X$ is a simple $R$-module $iff$ For any $x , y in X$, $x,y neq 0$ , there exists $r in R$ such that $rx = y$.
Here $R$ is a commutative ring. Furthermore, by a simple $R$-module I mean a non zero module that has non proper submodules.
I am very new to the study of modules so any push in the right direction would be helpful.

Mark · Accepted Answer

$(a)to (b)$: Let $0ne x,yin X$. Then $Rx={rx: rin R}$ is a submodule of $X$. This submodule is nonzero, as it contains the element $x$. Since $X$ is simple it follows that $Rx=X$, and in particular $yin Rx$.
$(b)to (a)$: Let $Y$ be a nonzero submodule of $X$. So take any $0ne yin Y$. We will show that $Y=X$. Let $xin X$. By assumption there is some $rin R$ such that $x=ry$ (if $x=0$ you can take $r=0$), so in particular $xin Y$.

Simple $R$-module equivalent statement

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