In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 has a non-zero socle

Socle (mathematics)

-Socle of a group:In the context of group theory, the socle of a group G, denoted Soc, is the subgroup generated by the minimal non-trivial normal subgroups of G. The socle is a direct product of minimal normal subgroups...

, or equivalently if the Loewy length of every module is defined. The concepts are named after Alfred Loewy

Alfred Loewy

Alfred Loewy was a German mathematician who worked on representation theory. Loewy rings, Loewy length, and Loewy series are named after him.-References:...

.

Loewy length

The Loewy length and Loewy series were introduced by

If M is a module, then define the Loewy series M_α for ordinals

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

α by M₀ = 0, M_α+1/M_α = socle M/M_α, M_α = ∪_λ<α M_λ if α is a limit ordinal. The Loewy length of M is defined to be the smallest α with M = M_α, if it exists.