The most straightforward graded version of the Koethe conjecture would seem to say that the result of summing of two graded-nil ideals produces a graded-nil ideal. Here, graded-nil means having all homogeneous elements nilpotent. The case of rings graded by the natural numbers interests me most, but I'd like to learn whatever I can.
Does this occur in the literature? I haven't spotted it myself in papers and books on radicals of graded rings, but among other things, I could have easily missed an equivalent formulation.
Does the original Koethe conjecture imply this more general version (which trivially contains the original by giving every element degree 0)?
Or is this more general version actually known to be false?
