Finitely generated submodule of a module [closed]

Let $M$ be an (infinitely generated) free module over a ring $R$, with generating set(not basis) $\{\alpha_i\}_{i \in \Lambda}$ . Let $N$ be a finitely generated $R$-module such that $f: N \rightarrow M$ is an injective module homomorphism. Can we say that there are finitely many $\{\alpha_i\}$ such that image of $N$ is contained in the submodule generated by them?

closed as off-topic by Strants, Mostafa Ayaz, José Carlos Santos, Xander Henderson, TaroccoesbroccoJun 27 '18 at 5:59

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Strants, Mostafa Ayaz, José Carlos Santos, Xander Henderson, Taroccoesbrocco
If this question can be reworded to fit the rules in the help center, please edit the question.

• write down the image of generators of $N$, see what happens – chí trung châu Jun 26 '18 at 15:45
• @AlexFrancisco Thanks, your comment forced me to write the idea in mind rigorously. I basically wanted to make sure there isn't some sort of flaw in that. – m_ath Jun 26 '18 at 16:02

Sure, if $\{\beta_j\}_{j=1}^n$ is the finite set that generates $f(N)$ (this set there exists because $f$ restrict to image is an isomorphism and $N$ is finitely generated) than $\beta_j=\sum_{k=1}^{n_j}a_{jk}\alpha_{kj}$ for each $j\in \{1,...,n\}$ and so $f(N)$ is contained by the module generated by $\{\alpha_{kj}: k=1,...n_j, j=1,...n\}$ that is a finite set of $\{\alpha_i\}_{i\in \Lambda}$
Take $\{\beta_j\}$ a finite generating set for $N$. Since $N\subseteq M$, each $\beta_j$ can be written in terms of finitely many $\alpha_i$. Take the collection of all those $\alpha_i$ appearing in the expressions of the $\beta_j$.