mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a left (right) Loewy ring or left (right) semi-Artinian ring is a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

in which every non-

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

left (right) module has a non-zero socle, or equivalently if the Loewy length of every left (right) module is defined. The concepts are named after Alfred Loewy.

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 α by ''M''₀ = 0, ''M''_α+1/''M''_α = socle(''M''/''M''_α), and ''M''_α = ∪_λ<α ''M''_λ if α is a

limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...

. The Loewy length of ''M'' is defined to be the smallest α with ''M'' = ''M''_α, if it exists.

Semiartinian modules

_R M

is a semiartinian module if, for all

epimorphism In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analo ...

M \rightarrow N

, where

N \neq 0

, the socle of

N

is essential in

N.

Note that if

_R M

is an

artinian module Artinian may refer to: Mathematics *Objects named for Austrian mathematician Emil Artin (1898–1962) **Artinian ideal, an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0 **Artinian ring, a ring which satisfies ...

then

_R M

is a semiartinian module. Clearly 0 is semiartinian. If

0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0

is exact then

M'

and

M''

are semiartinian if and only if

M

is semiartinian. If

\_

is a family of

R

-modules, then

\oplus_M_

is semiartinian if and only if

M_j

is semiartinian for all

j \in I.

Semiartinian rings

R

is called left semiartinian if

_R

is semiartinian, that is,

R

is left semiartinian if for any left ideal

I

R/I

contains a

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...

. Note that

R

left semiartinian does not imply that

R

is left artinian.

References

* * * *{{Citation , last1=Nastasescu , first1=Constantin , last2=Popescu , first2=Nicolae , title=Sur la structure des objets de certaines catégories abéliennes , year=1966 , journal=Comptes Rendus de l'Académie des Sciences, Série A , volume=262 , pages=A1295-A1297, publisher= GAUTHIER-VILLARS/EDITIONS ELSEVIER 23 RUE LINOIS, 75015 PARIS, FRANCE Ring theory