A “going down” theorem for certain reflected radicals

Barry J. Gardner and Patrick Noble Stewart

Vol. 55 (1974), No. 2, 381–389

DOI: 10.2140/pjm.1974.55.381

Abstract

In a category 𝒦 suitable for radical theory, a functor Φ : 𝒦→𝒦 is studied which is associated with a natural transformation 1_𝒦 → Φ in a way which bears a formal resemblance to the behavior of certain “extension” functors of rings, such as that which assigns to each A the polynomial ring A[x]: every normal subobject N → Φ(A) has a “contraction” N^c → A. For a radical class ℛ in 𝒦 such that ℛ^∗ = {A|Φ(A) ∈ℛ} is also radical, some conditions are obtained which imply that ℛ^∗(A) = ℛ(Φ(A))^c.

Mathematical Subject Classification 2000

Primary: 18A99

Secondary: 16A21

Milestones

Received: 4 December 1973

Published: 1 December 1974