In
mathematics, a Loewy ring or semi-Artinian ring is a
ring in which every non-
zeromodule has a non-zero
socle, or equivalently if the Loewy length of every module is defined. The concepts are named after
Alfred Loewy.
If M is a module, then define the Loewy series Mα for
ordinals α by M0 = 0, Mα+1/Mα = socle(M/Mα), and Mα = ∪λ<αMλ if α is a
limit ordinal. The Loewy length of M is defined to be the smallest α with M = Mα, if it exists.
Semiartinian modules
is a semiartinian module if, for all
epimorphisms, where , the socle of is essential in
Note that if is an
artinian module then is a semiartinian module. Clearly 0 is semiartinian.
If is
exact then and are semiartinian if and only if is semiartinian.
If is a family of -modules, then is semiartinian if and only if is semiartinian for all
Semiartinian rings
is called left semiartinian if is semiartinian, that is, is left semiartinian if for any left
ideal, contains a
simplesubmodule.
Note that left semiartinian does not imply that is left artinian.
References
Assem, Ibrahim; Simson, Daniel; Skowroński, Andrzej (2006), Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory, London Mathematical Society Student Texts, vol. 65, Cambridge:
Cambridge University Press,
ISBN0-521-58631-3,
Zbl1092.16001