I'm sorry if this question is too basic for MO. I'm reading [a paper by Graham and Lehrer "Cellular algebras"](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002113384) and have trouble understanding one step in a proof of a crucial theorem. I suppose that the problem is probably quite easy, but for some reason I got totally stuck.


It is proposition 3.2 (ii) on page 11: Let $(\Lambda, M,C,\ast)$ be a cell datum for a cellular algebra $A$, $\lambda \in \Lambda$,$rad(\lambda) = \{ x \in W(\lambda) | \phi\_\lambda (x,y) = 0 \ \forall y \in W(\lambda) \}$. Then if $\phi\_\lambda \ne 0$, the quotient $W(\lambda) / rad(\lambda)$ is (absolutely) irreducible. 


It is shown in the course of the proof that $W(\lambda) = Az$ for some $z \notin rad(\lambda)$. Then it says "it follows that $W(\lambda) / rad(\lambda)$ is irreducible". 


Can someone explain why this implication holds?



Question about a proof in Graham and Lehrer's "Cellular algebras"
If $W(\lambda)=Az$ for all $z\notin rad(\lambda)$, then $rad(\lambda)$ is a maximal submodule of $W(\lambda)$ (in fact it follows that it is the *unique* maximal submodule and therefore it is equal to the module-theoretic radical of $W(\lambda)$). In particular $W(\lambda)/rad(\lambda)$ is irreducible. It follows from Corollary 2.6 that it is absolutely irreducible.


