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All our rings are commutative with unity.

For an $R$-module $M$ and a submodule $N$ of $M$ and ideal $I$ of $R$, let $(M:I):=\{m\in M : Im \subseteq M\}$$(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 $M=R$.

All our rings are commutative with unity.

For an $R$-module $M$ and ideal $I$ of $R$, let $(M:I):=\{m\in M : Im \subseteq M\}$. 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 $M=R$.

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 $M=R$.

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user111524

All our rings are commutative with unity.

For an $R$-module $M$ and ideal $I$ of $R$, let $(M:I):=\{m\in M : Im \subseteq M\}$. 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 (M+Rm)<\alpha$$\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 $M=R$.

All our rings are commutative with unity.

For an $R$-module $M$ and ideal $I$ of $R$, let $(M:I):=\{m\in M : Im \subseteq M\}$. 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 $\mu (M+Rm)<\alpha$ 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 $M=R$.

All our rings are commutative with unity.

For an $R$-module $M$ and ideal $I$ of $R$, let $(M:I):=\{m\in M : Im \subseteq M\}$. 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 $M=R$.

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user111524
user111524

On minimal generating sets of certain submodules

All our rings are commutative with unity.

For an $R$-module $M$ and ideal $I$ of $R$, let $(M:I):=\{m\in M : Im \subseteq M\}$. 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 $\mu (M+Rm)<\alpha$ 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 $M=R$.