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.

Principle

Starting from the socle and iterating formation of successive quotients by the radical (or dually iterating radicals), the Loewy series stratifies a module into semisimple layers; the Loewy length counts these layers and captures how far the module is from being semisimple.

Demonstration

Over the local Artin algebra k[x]/(x^n), the regular module has Loewy length n because successive powers of the radical (generated by x) produce n nonzero layers before reaching zero; a semisimple module has Loewy length 1.

Misapplication

Confusing Loewy length with composition length (which counts simple factors in a composition series) or using Loewy length without ensuring the module is Artinian (where the Loewy series might not stabilize).

Consequence

Finite Loewy length implies an Artinian (and often Noetherian) structure with a finite semisimple filtration; knowledge of the Loewy length constrains extension groups and homological dimensions and guides module classification.

Reversal

Infinite Loewy length corresponds to modules whose radical series does not terminate, indicating infinitely deep non-semisimple structure; the opposite extreme, Loewy length 1, is semisimplicity.

Boundary

Applies to modules over rings where radicals and socles are well-behaved, typically Artinian rings and finite-dimensional algebras; it excludes general modules over rings with infinite descending chains or without a finite radical series.

Semantic Tension

Competes with composition length and radical/nilpotency indices: Loewy length measures stratified semisimplicity while composition length counts simple factors; both relate but give different granular information.

Synthesis

Loewy length is the finite count of semisimple layers obtained by iterating socle/radical constructions, giving a depth measure of how a module decomposes into successive semisimple pieces.