  Utilizing the linking algebra of a Hilbert $C^*$-module $\big(\mathscr{V},
{\|\!\cdot\!\|}_{_{\mathscr{V}}}\big)$, we introduce $\Omega(x)$ as a
definition of numerical radius for an element $x\in\mathscr{V}$ and then show
that $\Omega(\cdot)$ is a norm on $\mathscr{V}$ such that
$\frac{1}{2}{\|x\|}_{_{\mathscr{V}}} \leq \Omega(x) \leq
{\|x\|}_{_{\mathscr{V}}}$. In addition, we obtain an equivalent condition for
$\Omega(x) = \frac{1}{2}{\|x\|}_{_{\mathscr{V}}}$. Moreover, we present a
refinement of the triangle inequality for the norm $\Omega(\cdot)$. Some other
related results are also discussed.
