Consider the Cellular Algebra $A$ with a poset $\Lambda$, for any $\lambda,\mu$ in $\Lambda$, there is a A-module $C^\lambda$ and a irreducible A-module $D^\mu=C^\mu/radC^\mu$.
Denote $d_{\lambda\mu}$ the composition multiplicity of the irreducible A-module $D^\mu$ in $C^\lambda$.
I can not understand "$d_{\lambda\mu}\neq0$ if and only if there exists a submodule $M$ of $C^\lambda$ and a homomorphism $\theta:C^\mu\longrightarrow C^\lambda/M$ with the image of $\theta$ is isomorphic to $D^\mu$"..
(This is page 13, from Andrew Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group(1998).)
Thanks everyone!