On one-dimensional socles

Question

Let $(R,m,k)$ be a regular local ring of dimension $n$. Let $b_1,\dots,b_n$ be a maximal $R$-sequence and define $J=(b_1,\dots,b_n)$. Let $y_1,\dots,y_n$ be a regular system of parameters of $R$ and write $b_i = \sum b_{ji} y_j$. Then the $\operatorname{Hom}_R (k, R/J) \cong \det(b_{ji}) R/J$ (Corollary 2.3.10 in Bruns and Herzog, CMR).

Question: Why is it true that $\det(b_{ji})$ is inside every ideal $J'$ of $R$ such that $J' \supset J$?

Reference: Bruns and Herzog, CMR, second paragraph of proof of Theorem 2.3.16.

Answer 1

Let $(S,\mathfrak m)$ be a local artinian ring with socle $(0:\mathfrak m)=aR$, $a\ne 0$. Then $aR\subseteq I$ for any ideal $I\ne (0)$.

If there is an $x\in I$, $x\ne 0$, such that $x\in aR$, then $x=au$ with $u$ invertible (otherwise $u\in\mathfrak m$ and $x\in a\mathfrak m=0$), so $a\in I$.

If for any $x\in I$, $x\ne 0$, we have $x\notin aR$ then for any $x\in I$, $x\ne 0$, there is $m\in\mathfrak m$ such that $xm\ne 0$. For a fixed such $x$ choose $m_1\in\mathfrak m$ such that $m_1x\ne 0$. Repeat the argument for $m_1x$ and find $m_2\in\mathfrak m$ such that $m_2m_1x\ne 0$, and so on. On the other side, $\mathfrak m$ is nilpotent, a contradiction!