  We investigate the quotient algebra $\mathfrak{A}_X^{\mathcal I}:=\mathcal
I(X)/\overline{\mathcal F(X)}^{||\cdot||_{\mathcal I}}$ for Banach operator
ideals $\mathcal I$ contained in the ideal of the compact operators, where $X$
is a Banach space that fails the $\mathcal I$-approximation property. The main
results concern the nilpotent quotient algebras $\mathfrak
A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for the quasi
$p$-nuclear operators $\mathcal{QN}_p$ and the Sinha-Karn $p$-compact operators
$\mathcal{SK}_p$. The results include the following: (i) if $X$ has cotype 2,
then $\mathfrak A_X^{\mathcal{QN}_p}=\{0\}$ for every $p\ge 1$; (ii) if $X^*$
has cotype 2, then $\mathfrak A_X^{\mathcal{SK}_p}=\{0\}$ for every $p\ge 1$;
(iii) the exact upper bound of the index of nilpotency of $\mathfrak
A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for $p\neq 2$ is
$\max\{2,\left \lceil p/2 \right \rceil\}$, where $\left \lceil p/2 \right
\rceil$ denotes the smallest $n\in\mathbb N$ such that $n\ge p/2$; (iv) for
every $p>2$ there is a closed subspace $X\subset c_0$ such that both $\mathfrak
A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ contain a countably
infinite decreasing chain of closed ideals. In addition, our methods yield a
closed subspace $X\subset c_0$ such that the compact-by-approximable algebra
$\mathfrak A_X=\mathcal K(X)/\mathcal A(X)$ contains two incomparable countably
infinite chains of nilpotent closed ideals.
