A question about $rad(M)$

Question

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$?

Thanks!

Answer 1

Yes. A submodule of a small submodule is small and the sum of two small submodules is small.

A submodule $L$ of $M$ is small if, for all submodules $X$ of $M$, if $L+X=M$ then $X=M$.

If $L'$ is a submodule of $L$ and $L$ is small, then from $L'+X=M$ it follows $L+X=M$ and so $X=M$.

If $L_1$ and $L_2$ are small and $(L_1+L_2)+X=M$, we have $L_1+(L_2+X)=M$, so $L_2+X=M$ and so $X=M$.

Now, if $x\in\operatorname{rad}(M)$, then $x=\sum_{i=1}^n x_i$, with $x_i\in L_i$, where $L_i$ is small in $M$. But then $x\in L=L_1+\dots+L_n$, and $L$ is small. Therefore $Rx$ is a submodule of $L$, so it is small.

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Note, however, that $\operatorname{rad}(M)$ need not be small; a counterexample is the Prüfer $p$-group as a $\mathbb{Z}$-module, which is equal to its radical, because each proper submodule is small.