Ideal ring extensions and trusses

Abstract

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.