A ring with identity is called a clean ring if every element of the ring decomposes as a sum of an idempotent and a unit of the ring.

Trying to answer an open question put on by W. K. Nicholson, we introduce and investigate the new concept of a nice module, that is a module whose every submodule has a clean endomorphism ring. We present some sufficient conditions under which a module is nice, give examples of module that are nice and those that are not nice, show how to construct nice modules and raise an interesting question in the meantime.

clean module, automorphism invariant module, module of finite length.