The classical result of Gohberg, Markus and Feldman states that, when $E$ is one of the classical Banach sequence spaces $E=l^p$ for $1\leq p<\infty$ or $E=c_0$, the only closed, two-sided, non-trivial ideal in $\cal B(E)$, the Banach algebra of operators on a Banach space $E$, is $\cal K(E)$, the ideal of compact operators. Gramsch and Luft completely classified the closed, two-sided ideals in $\cal B(H)$ for an arbitrary Hilbert space $H$ through the idea of $\kappa$-compact operators, for infinite cardinals $\kappa$. This paper presents an extension of this result to the non-separable versions of $l^p$ and $c_0$.