Masas in quotients

Masas in quotients

Let $A$ be a von Neumann algebra and let $B$ be a norm-closed ideal of $A$
(but not necessarily WOT-closed). What one has to assume about $A$ and $B$
to ensure that if $M\subset A$ is a maximal abelian subalgebra, then $M /
(B\cap M)$ is a maximal algebian subalgebra of $A/B$? This is the case for
$A=B(H)$ and $B=K(H)$.