  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.

Rings and Algebras