  It is shown that there is a close relationship between ideal extensions of
rings and trusses, that is, sets with a semigroup operation distributing over a
ternary abelian heap operation. Specifically, a truss can be associated to
every element of an extension ring that projects down to an idempotent in the
extending ring; every weak equivalence of extensions yields an isomorphism of
corresponding trusses. Furthermore, equivalence classes of ideal extensions of
rings by integers are in one-to-one correspondence with associated trusses up
to isomorphism given by a translation. Conversely, to any truss $T$ and an
element of this truss one can associate a ring and its extension by integers in
which $T$ is embedded as a truss. Consequently any truss can be understood as
arising from an ideal extension by integers. The key role is played by
interpretation of ideal extensions by integers as extensions defined by double
homothetisms of Redei [L.\ Redei, Die Verallgemeinerung der Schreierschen
Erweiterungstheorie, {\em Acta Sci.\ Math.\ Szeged}, {\bf 14} (1952), 252--273]
or by self-permutable bimultiplications of Mac Lane [S.\ Mac Lane, Extensions
and obstructions for rings, {\em Illinois J.\ Math.} {\bf 2} (1958), 316--345],
that is, as {\em integral homothetic extensions}. It is shown that integral
homothetic extensions of trusses are universal as extensions of trusses to
rings but still enjoy a particular smallness property: they do not contain any
subrings to which the truss inclusion map corestricts. Minimal extensions of
trusses into rings are defined. The correspondence between homothetic ring
extensions and trusses is used to classify fully up to isomorphism trusses
arising from rings with zero multiplication and rings with trivial
annihilators.
