CORRECTION TO THE ARTICLE ON THE EMBEDDED ASSOCIATED PRIMES OF MONOMIAL IDEALS

Abstract

Let $I\subset R=K[x_1,\dots ,x_n]$ be a monomial ideal, $\mathfrak{𝔪}=(x_1,\dots ,x_n)$, $t$ a positive integer, and $y_1,\dots ,y_s$ be distinct variables in $R$ such that, for each $i=1,\dots ,s$, $\mathfrak{𝔪}\setminus y_i\notin \mathrm{Ass}(R∕{(I\setminus y_i)}^t)$, where $I\setminus y_i$ denotes the deletion of $I$ at $y_i$. It is shown in Theorem 3.4 of the article in question that $\mathfrak{𝔪}\in \mathrm{Ass}(R∕I^t)$ if and only if $\mathfrak{𝔪}\in \mathrm{Ass}\left(R∕\left(I^t:{\prod }_{i=1}^s{y}_i\right)\right)$. As an application of Theorem 3.4, it is argued in Theorem 3.6 that under certain conditions, every unmixed König ideal is normally torsion-free. In addition, Theorem 3.7 states that under certain conditions a square-free monomial ideal is normally torsion-free. It turns out that these conditions are not enough to obtain the desired statements in Theorems 3.6 and 3.7. We update these conditions to validate the conclusions of Theorems 3.6 and 3.7. For this purpose, it is enough for us to replace the expression “$\mathfrak{𝔪}\setminus x_i\notin \mathrm{Ass}(R∕{(I\setminus x_i)}^t)$” with the new expression “$I\setminus x_i$ is normally torsion-free”. It should be noted that the previous proofs are still correct.