Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:_Rx)}\}.$$ If $R$ or $M$ is noetherian, then this submodule has lots of nice properties, some of which are lost in a non-noetherian setting. While trying to understand what precisely is lost, I got stuck with the following question:

Is there an example of $R$, $\mathfrak{a}$ and $M$ such that:

 

(1) There exists $n\in\mathbb{N}$ with $\mathfrak{a}^n\Gamma_{\mathfrak{a}}(M)=0$;

 

(2) For every $m\in\mathbb{N}$ we have $\mathfrak{a}^mM\cap\Gamma_{\mathfrak{a}}(M)\neq 0$.

One should note that (1) implies $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^nx=0\}$$ (which is not true in general).

Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:_Rx)}\}.$$ If $R$ or $M$ is noetherian, then this submodule has lots of nice properties, some of which are lost in a non-noetherian setting. While trying to understand what precisely is lost, I got stuck with the following question:

Is there an example of $R$, $\mathfrak{a}$ and $M$ such that:

(1) There exists $n\in\mathbb{N}$ with $\mathfrak{a}^n\Gamma_{\mathfrak{a}}(M)=0$;

(2) For every $m\in\mathbb{N}$ we have $\mathfrak{a}^mM\cap\Gamma_{\mathfrak{a}}(M)\neq 0$.

One should note that (1) implies $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^nx=0\}$$ (which is not true in general).