Suppose $R$ is a right noetherian ring. Prove that every onesided nilideal is nilpotent.
I try to use this theorem: If R is a commutative Ring and I is nilideal of R and also I is finitely generated, then I is nilpotent.
Because R is right noetherian then R is noetherian as a right R-module, so I. Conclude that (by one theorem) every submodule of R will be finitely generated, but for using the theorem I mentioned above, R must be commutative, so I could not go farther because there is not this hypothesis.
A suggestion or a hint will be usefull for me.