faithful $R$-module same as $R$ acting faithfully?

Question

This is a rather simple question. Do we mean the same thing when we say we have a faithful $R$-module and the ring $R$ acts faithfully on an $R$-module $M$? I've been having trouble finding clarification and I suspect the terms are interchangeable but I am aware that a faithful group action is defined a bit differently so I want to be careful.

Answer 1

That's correct. The phrases "faithful $R$ module $M$ and "$R$ acts faithfully on $M$" mean the same thing.

Answer 2

It is the same thing.

$M$ is a faithful $R$-module if $\forall r \in $R$, \exists m \in M : rm \neq 0$. The action of $R$ on $M$ is faithful if $R \to \mathrm{End}_R(M), r \mapsto (x \mapsto rx)$ is injective.