On minimal generating sets of certain submodules

Question

All our rings are commutative with unity.

For an $R$-module $M$ and a submodule $N$ of $M$ and ideal $I$ of $R$, let $(N:I):=\{m\in M : Im \subseteq N\}$. Let $\mu (M)$ denote the least cardinality among the generating sets of $M$.

Now let $\alpha$ be an infinite cardinal. Let $M$ be a faithful $R$-module such that $\mu(M) <\alpha$. If $r\in R ,m \in M$ and $N$ is a submodule of $M$ such that $rm \in N$ and $\mu (N+Rm)<\alpha$ and $\mu ((N: Rr)) <\alpha$, then is it true that $\mu (N) <\alpha$ ? If this is not true for every infinite cardinal $\alpha$, then for which $\alpha$ is it true ?

I can show that the assertion is true when $M=R$.

Answer 1

Are you missing some conditions? I think the following is a counterexample with $M=R$.

Let $k$ be a field and $R=k[x_i\mid i\in I]$, with $|I|=\alpha$, a polynomial ring in $\alpha$ many variables.

Take $M=R$, $N=\langle x_i\mid i\in I\rangle$, $m=1$, and $r$ any element of $N$.

Then

• $\mu(M)=1$,
• $\mu(N)=\alpha$,
• $rm=r\in N$
• $N+Rm=R$, so $\mu(N+Rm)=1$,
• $(N:Rr)=R$, so $\mu\left((N:Rr)\right)=1$.