Jordan-Holder vs Harder-Narasimhan 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
 A: If your algebra is finite-dimensional, then this is the Loewy-Filtration and explicitly $F^{k}M = J(A)^k M$ where $J(A)$ is the Jacobson-radical of $A$.
Also note: Without any finiteness condition your filtration is not well-defined. Consider for example $A=K[X]$ for some field $X$ and choose pairwise distrinct, monic, and irreducible polynomials $p_k$. The Ideals $(p_1) \supseteq (p_1 p_2) \supseteq (p_1 p_2 p_3) \supseteq \ldots$ all have semi-simple quotients by the chinese remainder theorem. Their intersection is $0$ so that if $F^1(A)$ exists, it must necessarily be zero. On the other hand $F^0 M/F^1 M=A$ is not semisimple so that there is no smallest submodule with semisimple quotient (not even a minimal one).
A: 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").
