On congruence classes and extensions of rings with applications to braces

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.