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

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.

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.
• It is not true in general that $R\rightarrow R'$ is faithfully flat and it is not assumed in the prerequesite of the exercise. Also, $M$ in the first part is supposed to be an arbitrary $R$ - module, so $(M\otimes_R R')_{/R}$ will not even be flat. Mar 17, 2020 at 17:18
• The exercise says that the restrictions of $R'$-modules are faithfully flat over $R$, does it not?
• The way I see it, it says that we have a given $R'$ - module $M'$ such that its restriciton $M'_{/R}$ on $R$ via $R\rightarrow R'$ is faithfully flat. Though, I think in order for the exercise to give valid statements, we should assume something about $R\rightarrow R'$ or rephrase the exercises... Mar 18, 2020 at 15:12