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When is the $I^{\prime }$ a complete intersection?

Definition (section 2.3 of
Let $S$ be a Noetherian local ring and $I$ and $I^{\prime }$ ideals in $S$. We say $I^{\prime }$ is a complete intersection if $S$ has a free presentation:
$$I^{\prime } \cong S^{\oplus r} \oplus J \oplus K,$$
and $J$ and $K$ are both generated by $r-1$ elements.
Moreover, we say $S$ has a regular element if $S$ has a regular element that is not in the ideal $(0)$.

Do we know when $I^{\prime }$ is a complete intersection? When can one use a regular element to compute $I^{\prime }$? Is this question even well-defined?


The ring $S$ always has regular elements, one for each maximal ideal, and since no maximal ideal contains a regular element, all maximal ideals have the same regular elements, and the question has an unambiguous answer.

1. Field of the Invention
The present invention relates to a lens driver circuit. More specifically, the present invention relates to a lens driver circuit for driving a movable lens of an optical apparatus such as a still camera, a video camera or a telescope.
2. Description of the Related Art
A lens driver circuit is disposed in a lens barrel of an optical apparatus such as a still camera or a video camera. The lens driver circuit drives a movable lens of the optical apparatus to move in a direction parallel to the optical axis of the optical apparatus so as to vary the focal distance.
A lens driver circuit of the type referred to above is disclosed in, e.g., Japanese Patent Laid