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Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

Let $G=K_{n_1,n_2,...,n_r}$ be a complete multipartite graph on $\left[n\right]$ with $n>r>1$ and $J_G$ being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity $\mathrm{reg}\left(J_G^t\right)$ is $2t+1$ for any positive integer $t$.

Let $(R,𝔪)$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq {(x_1,...,x_n)}^2$ is a graded ideal of a polynomial ring $S=K[x_1,...,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ of the first syzygy module ${\mathrm{Syz}}_1\left(𝔪\right)$ of $𝔪$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.