Loewy Length

The number of nonzero layers in the Loewy (radical) series of a module or algebra: the least integer n such that the n-th term of the Loewy series equals the whole module, measuring the depth of successive radical quotients until a semisimple object is reached.

The Loewy length of a module M (or algebra regarded as a module over itself) is the smallest ordinal n (often a positive integer) such that the Loewy (socle) filtration reaches M after n steps: 0 = L_0 ⊂ L_1 ⊂ ··· ⊂ L_n = M where each successive quotient L_{i+1}/L_i is the socle of M/L_i.