Adjoining an Identity to a Reduced Archimedean f-Ring

Abstract

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding u _A :A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (u _A , uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are “u-extendable” is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ∀ B(frA(A, B) = U (A, B)), and, ∀ B ≠ 0(frA(B, A) = U(B, A)).

Keywords:

• Absolute extensors
• Archimedean
• Extended real-valued function
• f-Ring
• Reflective subcategory

2000 Mathematics Subject Classification:

• Primary 06F25, 54H10
• Secondary 13B10, 18A40, 46A40, 54C30

ACKNOWLEDGMENT

The authors wish to thank the referee for a perspicacious reading of the manuscript and the resulting suggestions that have led to noticeable improvement of the final product.

Notes

Communicated by K. M. Rangaswamy.