Ideal ring extensions and trusses (Q2118943)

An abelian heap is a set \(H\) with a ternary \([\_,\_,\_]\) operation on \(H\). This ternary operation is subject to certain requirements which ensure that for any \(e\in H,\) the binary operation \(a+_{e}b=[a,e,b]\) gives an abelian group with identity \(e\) on \(H.\) Conversely, any abelian group \(G\) determines an abelian heap by defining \([a,b,c]=a-b+c.\) An abelian heap endowed with an associative binary multiplication that distributes from the left and the right over the ternary operation is called a truss. A truss equipped with a specific nullary operation or with an element with special properties can be made into a ring. Conversely, every ring can be made into a truss in a natural way, by associating the (unique) heap operation \([a,b,c]=a-b+c\) to the abelian group operation. The structure of trusses and their relationships to other algebraic structures, for example braces, have been investigated during the last few years. Here the authors study the relationship between ideal extensions of rings and trusses. The main results show that a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring, while every weak equivalence of extensions yields an isomorphism of corresponding trusses. Conversely, to any truss and an element of this truss one can associate a ring together with its extension by integers in which the truss is embedded as a truss. In dealing with the ring extensions, Redei's theory of homothetisms and Mac Lane's self-permutable bimultiplications play an important role. Any infinite homothetic extension is equivalent to an extension by the ring of integers. A categorical interpretation of the infinite homothetic ring extensions \(T(e)\) in which a truss \(T\) embeds is given and it is shown that it has the universal property that any truss homomorphism from \(T\) to a ring \(R\) factorises through the inclusion \(T\rightarrow T(e)\) and a unique ring homomorphism \(T(e)\rightarrow R\). Several applications of the theory are given. A classification or description of all trusses induced by homothetic data on rings with zero multiplication and on rings with trivial annihilator is given. In conclusion, a full classification (up to isomorphism) of trusses that can be constructed on the heap determined by the abelian group \(\mathbb{Z}_{p}\oplus \mathbb{Z}_{p}\) for a prime \(p\), is provided.