Quasi-socle ideals in Buchsbaum rings

Abstract

Quasi-socle ideals, that is, ideals of the form $I=Q:{\mathfrak{m}}^q$ $(q\ge 2)$, with $Q$ parameter ideals in a Buchsbaum local ring $(A,\mathfrak{m})$, are explored in connection to the question of when $I$ is integral over $Q$ and when the associated graded ring $\mathrm{G}\left(I\right)={\oplus }_{n\ge 0}{I}^n/I^{n+1}$ of $I$ is Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after necessary modifications of the conditions on parameter ideals $Q$ and integers $q$. Examples are explored.