Let $C$ be an abelian ring, $\mathfrak{c}$ an ideal of $C$ and $E$ a $C$-module. Let $\mathfrak{c}E$ be the sub-$\mathbf{Z}$-module of $E$ generated by the family $(c x)_{(c,x)\in\mathfrak{c}\times E}$ of elements of $E$: i.e. $$\mathfrak{c}E=\left\{y\ |\ (\exists\mu)\left(\mu\in\mathbf{Z}^{(\mathfrak{c}\times E)}\ \land\ y=\sum_{(c,x)\in\mathfrak{c}\times E}\mu_{cx}(cx)\right)\right\}.$$
I would like to show that $\mathfrak{c}E$ is stable under the action of $C$. Let $\lambda\in C$ and $y\in\mathfrak{c}E$: there exists $\mu\in\mathbf{Z}^{(\mathfrak{c}\times E)}$ such that $y=\sum_{(c,x)\in\mathfrak{c}\times E}\mu_{cx}(cx)$. Then $$\lambda y=\lambda\left(\sum_{(c,x)\in\mathfrak{c}\times E}\mu_{cx}(cx)\right)=\sum_{(c,x)\in\mathfrak{c}\times E}\lambda\mu_{cx}(cx)=\sum_{(c,x)\in\mathfrak{c}\times E}\mu_{cx}\lambda(cx)=\sum_{(c,x)\in\mathfrak{c}\times E}\mu_{cx}(\lambda c)x.$$ Obviously $\lambda c\in\mathfrak{c}$ for $c\in\mathfrak{c}$. I am not able to rewrite the sum $\sum_{(c,x)\in\mathfrak{c}\times E}\mu_{cx}(\lambda c)x$ in a form that will allow me to conclude that $\lambda y\in\mathfrak{c}E$. What am I doing wrong?