On congruence classes and extensions of rings with applications to braces

TL;DR

This paper explores the deep connections between ring theory, trusses, and braces, showing how quotient structures and extensions relate to these algebraic objects, revealing new insights into their inherent relationships.

Contribution

It demonstrates that quotient classes in rings are paragon elements in trusses and describes how extensions of trusses by modules relate to braces, highlighting their structural interplay.

Findings

Quotient classes in rings are paragon elements in associated trusses.

Extensions of trusses by modules can produce braces.

Extended trusses associated to braces remain braces regardless of the module used.

Abstract

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring $R$ by an ideal $I$ is a paragon in the truss $\mathrm{T}(R)$ associated to $R$. Second, an extension of a truss by a one-sided module is described. Even if the extended truss is associated to a ring, the resulting object is a truss, never a ring, unless the module is trivial. On the other hand, if the extended truss is associated to a brace, the resulting truss is also associated to a brace, irrespective of the module used.