Natural map $M\rightarrow M\otimes_R R'$ injective - Missing prerequisite in Bosch's book?

Question

I am working my way through Bosch's book "Algebraic Geometry and Commutative Algebra" and found an exercise in chapter 4.4, exercise 3, which I think is missing some prerequisite. The original text is as follows:

Consider a ring morphism $R\rightarrow R'$, an $R'$-module $M'$, as well as its restriction $M'_{/ R}$ on $R$, and assume that the latter is a faithfully flat $R$-module. Show:

1. For any $R$ - module $M$, the canonical map $M\rightarrow M\otimes_R R'$, $x\mapsto x\otimes 1$ is injective.
2. Any ideal $\mathcal{I}\subset R$ satisfies $\mathcal{I}R'\cap R=\mathcal{I}$.
3. If $R'$ is Noetherian (Artinian), the same is true for $R$.

I think there must be some condition imposed on $R\rightarrow R'$ or $M'$ as an $R'$-module, because even the first exercise is only true for flat $R$ - modules $M$. The statements look rather standard, so I guess there is some well known condition.

Answer 1

I think I got it. The restriction $R'_{/R}$ is simply $R'$ with $r\cdot r'=f(r)r'$ for $r\in R, r'\in R'$, and this is faithfully flat. Now, the restriction $(M\otimes_RR')_{/R}$ is faitfully flat, so Bosch 4.4 Prop 1(iv) tells us that $(M\otimes_R R')_{/R}\otimes_R R'\cong M\otimes_R R'$ is faithfully flat. Same proposition but (v) tells us that $M$ is faithfully flat.