Quotients of Preradicals on a Module Category

• Published in: Communications in algebra Vol. 39; no. 8; pp. 2906 - 2925
• Main Authors: Teply, M. L., van den Berg, J. E.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01-08-2011
• Subjects: Algebra; Artinian module; Categories; Collection; Endofunctor; Hereditary preradical; Hereditary radical; Idempotent preradical; Loewy module; Loewy series; Mathematical analysis; Modules; Multiplication; Preradical; Primary 16S90; Quotients; Secondary 16E65, 16P20, 20M99; Semiartinian module; Semigroup; Semiprimary ring; Serial ring; Socle; Tables; Torsion; Torsion-free; Uniserial module; Wisbauer category
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927872.2011.582058

If σ and τ are preradical functors on the category of left modules over a ring R, it is possible to define a "quotient" endofunctor
by
for all modules M. The collection of all such endofunctors (denoted
is shown to constitute a semigroup under the operation of functor composition (modulo the congruence relation which identifies two functors ε, δ if ε(M) ≅ δ(M) for all modules M). Multiplication tables for this semigroup are constructed in some simple cases. An important subsemigroup of
is identified, and several important classes of rings are characterized in terms of
.