  We construct various examples of non-trivial closed ideals of the
compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$
on Banach spaces $X$ failing the approximation property. The examples include
the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X
\neq \{0\}$ and $\mathfrak{A}_Y \neq \{0\}$, then $\mathfrak{A}_{X \oplus Y}$
has at least $2$ closed ideals, (ii) there are closed subspaces $X \subset
\ell^p$ for $4 < p < \infty$ and $X \subset c_0$ such that $\mathfrak{A}_X$
contains a non-trivial closed ideal, (iii) there is a Banach space $Z$ such
that $\mathfrak{A}_Z$ contains an uncountable lattice of closed ideal having
the reverse order structure of the power set of the natural numbers. Some of
our examples involve non-classical approximation properties associated to
various Banach operator ideals. We also discuss the existence of compact
non-approximable operators $X \to Y$, where $X \subset \ell^p$ and $Y \subset
\ell^q$ are closed subspaces for $p \neq q$.
