### A note on the multiplier ideals of monomial ideals

Let $\U0001d51e\subseteq \u2102[{x}_{1},...,{x}_{n}]$ be a monomial ideal and $\mathcal{J}\left({\U0001d51e}^{c}\right)$ the multiplier ideal of $\U0001d51e$ with coefficient $c$. Then $\mathcal{J}\left({\U0001d51e}^{c}\right)$ is also a monomial ideal of $\u2102[{x}_{1},...,{x}_{n}]$, and the equality $\mathcal{J}\left({\U0001d51e}^{c}\right)=\U0001d51e$ implies that $0<c<n+1$. We mainly discuss the problem when $\mathcal{J}\left(\U0001d51e\right)=\U0001d51e$ or $\mathcal{J}\left({\U0001d51e}^{n+1-\epsilon}\right)=\U0001d51e$ for all $0<\epsilon <1$. It is proved that if $\mathcal{J}\left(\U0001d51e\right)=\U0001d51e$ then $\U0001d51e$ is principal, and if $\mathcal{J}\left({\U0001d51e}^{n+1-\epsilon}\right)=\U0001d51e$ holds for all $0<\epsilon <1$ then $\U0001d51e=({x}_{1},...,{x}_{n})$. One global result is also obtained. Let $\tilde{\U0001d51e}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $\U0001d51e$. Then it is proved that the equality $\mathcal{J}\left(\tilde{\U0001d51e}\right)=\tilde{\U0001d51e}$ implies that $\tilde{\U0001d51e}$ is principal.