Prove that $f_a:M\to M,m\mapsto ma$ is an isomorphism whenever it injective iff there is nonzero linear map $f:Ε\to Μ$ with $Ε$ injective.

Question

Prove that the following are equivalent for a commutative noetherian ring with identity:

(a) For every nonzero $R$-module $Μ$ there is a nonzero linear map $f:Ε\to Μ$ with $Ε$ injective.

(b) For every nonzero $R$-module $Μ$ and any $a\in R$, the endomorphism $f_a:M\to M,m\mapsto ma$ is an isomorphism whenever it is injective.

My attempt is:

$(b)\Rightarrow(a)$ By [Theorem 2.4.3] in the book Flat Covers of Modules by Jinzhong Xu, for every nonzero $R$-module $Μ$ there is a linear map $f:Ε\to Μ$ with $Ε$ injective.

I only have to prove that the map $f$ is non-zero. That is where I need help.