Vtyjkyuk

I'm studying Morita theorem. I have the following property:

Let $R$ and $S$ Morita equivalent via $F: Rtext-mathrmmod to Stext-mathrmmod$ and suppose $M$ is an $R$-module. Then $M$ is finitely generated if and only if $F(M)$ is finitely generated.

I use the following characterization of finitely generated modules: $M$ is finitely generated if and only if, given a family $N_i mid i in I$ of submodules of $M$ such that $M=sum_i in IN_i$ there exists a finite set $J subset I$ such that $M=sum_iin J N_i$.

Now I consider a family $N_i mid i in I$ of submodules of $F(M)$ such that $sum_i in IN_i=F(M)$. There exist $A_i$ such that $F(A_i)=N_i$ for every $i in I$. The author says that $A_i$ are $R$-modules. So $F(M)=F(sum_i in IA_i)$. Then, if I apply $G$ to both sides I have $M= sum_i in IA_i$. I use the fact that $M$ is finitely generated to find a subset $J$ of $I$ and I can conclude that $F(M)$ is finitely generated.

My doubt is when he says that $A_i$ is an $R$-module. It is true, but I need the $A_i$ to be submodules of $M$ right? So I can apply the characterization of finitely generated modules. Is there a way to prove that the $A_i$ are submodules of $M$ knowing that the $N_i$ are submodules of $F(M)$ and that $N_i=F(A_i)$? Thanks!

Let $G$ denote an inverse of $F$ with $G(N_i) = A_i$. Then $G$, being exact, sends the monomorphism $N_ihookrightarrow F(M)$ to the monomorphism $A_ihookrightarrow G(F(M))cong M$. Hence you can identify $A_i$ with a submodule of $M$