TITLE: Jordan-Holder vs Harder-Narasimhan
QUESTION [5 upvotes]: Let $M$ be a module over an algebra or a group. I am interested the following decreasing filtration: 
$F^0M=M$; 
$F^iM$ is the smallest sub-module of $F^{i-1}M$ such that the quotient is semi-simple. This filtration is unique.
Does it have a specific name?? Harder-Narasimhan?? Any reference?? 
(Jordan-Holder is when the quotients are simple, so it is a refinement of this one)
Thanks

REPLY [2 votes]: I usually refer to this as the "radical filtration," since the module $F^iM$ is the radical (or Jacobson radical) of the module $F^{i-1}M$.  While not explicitly stated, this paper seems to suggest it could also be called the "upper Loewy series" (with the lower Loewy series being the dual thing I would usually call the "socle filtration").