# Graded endomorphisms factoring through nilpotent ones

Let $$M$$ be a finitely generated 2-graded $$k[x,y]$$-module, concentrated in nonnegative degrees, and $$f$$ be a degree $$(n,m)$$-endomorphism of $$M$$. We can consider $$f$$ as a morphism $$M\langle n,m\rangle\to M_{\geq (n,m)}$$, where $$M_{\geq (n,m)}$$ is the submodule of $$M$$ in degrees $${}\geq (n,m)$$.

Definition: An element $$r\in R$$ of some ring is called strongly nilpotent if $$\forall s\in R$$ the element $$rs$$ is nilpotent.

Are the following two conditions equivalent?

1. $$f$$ factors through a strongly nilpotent degree $$(0,0)$$-endomorphism of $$M_{\geq (n,m)}$$,
2. $$f$$ factors through a strongly nilpotent degree $$(n,m)$$-endomorphism of $$_{\geq (n',m')}$$ for some $$(0,0)\leq (n',m')\leq (n,m)$$?

Of course, 1. implies 2., but how about the converse? If $$f$$ factors through a strongly nilpotent element for some $$(n', m')$$, then $$f=g'hg$$ for $$g$$ of degree $$(n,m)$$, $$h$$ of degree $$(0,0)$$ strongly nilpotent, and $$g'$$ of degree $$(n-n', m-m')$$.

Can anyone provide me with a counter example where 2. does not imply 1.?