##  [Loewy Length](/loewy-length) 

  ##  [Loewy Length](https://puremath.quantumdictionary.io/loewy-length-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

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.

 

 

 

 

 





 

 



 ##  [Loewy Length](https://algebra.quantumdictionary.io/loewy-length-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

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.