Definition: A linear topology $\tau$ on a left $A$-module $M$ is a topology on $M$ that is invariant under translations and admits a fundamental system of neighborhood of $0$ that consists of submodules of $M$.
From this definition, let $\mathcal{F}$ be a fundamental system of neighborhood of $0$ consisting of submodules of $M$. Since $\{0\}$ is a submodule of $M$, then $\{0\}\in\mathcal{F}$. Can we say that the linear topology on $M$ is a discrete topology? Because $\{0\}$ is in $\mathcal{F}$ implies $a+\{0\}=\{a\}$ is an open set in $M$.
Thank you in advance.