Quotients of preradicals on a module category

摘要

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 σ/τ by σ/τ(M) = σ(M)/τ(M) for all modules M. The collection of all such endofunctors (denoted Q(P)) is shown to constitute a semigroup under the operation of functor composition (modulo the congruence relation which identifies two functors ∈, δ if ∈(M) ? δ(M) for all modules M). Multiplication tables for this semigroup are constructed in some simple cases. An important subsemigroup of Q(P) is identified, and several important classes of rings are characterized in terms of Q (P).