Descent in the dual category of ternary rings

TL;DR

This paper investigates the structure of ternary rings, establishing unique normal forms for elements in free products and characterizing effective codescent morphisms, with implications for commutative rings.

Contribution

It demonstrates that the variety of ternary rings satisfies strong amalgamation and characterizes effective codescent morphisms within this category.

Findings

Elements in free products have unique normal forms.

The variety of ternary rings satisfies strong amalgamation.

Effective codescent morphisms are characterized in ternary rings and compared with those in commutative rings.

Abstract

It is shown that, in the variety of ternary rings, the elements of amalgamated free products have unique normal forms, and, moreover, this variety satisfies the strong amalgamation property. Applying these statements, effective codescent morphisms of ternary rings are characterized. In view of the fact that the category of ternary rings contains the category of commutative associative unitary rings as a full subcategory, the class of effective codescent morphisms in the latter category (which, according to the well-known Joyal-Tierney's criterion, are precisely monomorphisms $R\rightarrowtail S$ which are pure as monomorphisms of $R$-modules) is compared with that of morphisms between commutative associative unitary rings which are effective codescent in the category of ternary rings. It turns out that the former class is contained in the latter one, but does not coincide with it.