on $mathbb{z}_k-$vertex-magic labeling of prime graph $pg(mathbb{z}_n)$

Let $g=(v(g),e(g))$ be a graph, $(mathcal{a},+)$ be an abelian group with identity $0_{mathcal{a}}$, and $(mathcal{{r}},+,cdot)$ be a ring. the $mathcal{a}$-vertex-magic labeling of $g$ is a mapping from $v(g)$ to $mathcal{a}-{0_{mathcal{a}}}$ such that the total labels of every adjacent vertex with $u$ are equal for every $u$ in $v(g)$. the prime graph over ${mathcal{r}}$, denoted by $pg(mathcal{r})$, is a graph with $v(pg(mathcal{r}))={mathcal{r}}$ such that $uv$ is an edge if and only if $umathcal{r}v={0_{mathcal{r}}}$ or $v{mathcal{r}}u={0_{mathcal{r}}}$, for every vertex $uneq v$. in this article, we discuss the $mathbb{z}_k$-vertex-magic labeling of the prime graph over ther ring $mathbb{z}_n$. we study some literature to develop the properties of $mathbb{z}_k$-vertex-magic labeling of $pg(mathcal{r})$. we investigate some classes of prime graphs over ring $mathbb{z}_n$ for $n=p, n=p^2,$ and $n=pq$, with $pneq q$ primes.