  A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with
"free exterior Koszul homology"; the definition can also be understood in terms
of vanishing of Andr\'e-Quillen homology functors. Principal q.c.i. ideals are
well understood, but few constructions are known to produce q.c.i. ideals of
grade zero that are not principal. This paper examines the structure of q.c.i.
ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i.
ideal of $R$ is principal. On the other hand, we give an example of a minimal
q.c.i. deal $I$ which does not contain any principal q.c.i. ideals and is not
embedded, in the sense that no faithfully flat extension of $I$ can be written
as a quotient of complete intersection ideals. We also describe a generic
situation in which the maximal ideal of $R$ is an embedded q.c.i. ideal that
does not contain any principal q.c.i. ideals.
