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Let $I$ be a Cohen-Macaulay monomial ideal of $R=K[x_1,...,x_n]$, where $K$ is a field. Can we say the ideal $(x_1)+I$ is Cohen Macaulay?

Let $(R,m)$ be a regular local ring of dimension three and $I$ be an ideal of $R$. Is $\bar{I^{n+1}}=I^n\bar{I}$ for all positive integer $n$? where $\bar{I}$ is integral closure of $I$.

Let $J$ be a minimal reduction of $I$. By our assumption, we have $ht(J)\leq\mu(J)\leq\ell(I)\leq\mu(I)=ht(I)=ht(J)$. Thus $\mu(J)=\mu(I)$. From the exact sequence $$0\longrightarrow\frac{J}{mI\cap J …

Let $R=K[x_1,...,x_n]$ be the polynomial ring over a field $K$ and $I[x_1,...,x_n]=(u_1,...,u_t)$ be a Cohen-Macaulay monomial ideal of $R$. If $m<n$, could we say that $I[x_1,...,x_m,0,0,...,0]$ is …

Let $R$ be a commutative Noetherian ring and $I$, $J$ be two ideal of $R$. If $x\in R$, then is $((I+J):x)=(I:x)+(J:x)$? I would be very grateful if someone comment me.

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $R[It]$ be a rees ring of $R$ with respect to $I$. Do we have $Ass R[It]=\{pR[It] : p\in Ass(R)\}$? if not what can we say? ($Ass(R)$ is the a …

If $I$ and $J$ are monomial ideals, then the question is true.

Let $I$ and $J$ be two monomial ideals of $R=K[x_1,\dotsc,x_n]$ where $K$ is a field. Could we say $\mid\operatorname{depth} (R/I)- \operatorname{depth}(R/(I:J))\mid\leq 1$, where $(I:J)$ is colon ide …