About ring structures on the set of integers

D. Yu. Artemov

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 6-17

Résumé

It is well known that the ring of integers $\mathbb{Z}$ is an $E$-ring, therefore it is possible to define unique (up to isomorphism) structure of a ring with identity on the additive group $\mathbb{Z}$. A natural question arises about the uniqueness of the ring structure with identity constructed on a multiplicative monoid $\mathbb{Z}$. It is shown in this paper that this question is solved negatively. Moreover, a method of construction new various ring structures on the multiplicative monoid $\mathbb{Z}$ by dint of multiplicative automorphisms was developed and described. The concept of basis was introduced for the multiplicative monoid $\mathbb{Z}$, and it was shown that there are no bases (up to sign) that are differ to a basis consists of all prime numbers, and bases that are obtain of that basis by a permutations of its elements. The example of construction a new ring structure on the set $\mathbb{Z}$ for fixed standart multiplication is given in the end of this paper. The new addition on the multiplicative monoid $\mathbb{Z}$ is obtained by a permutation of prime numbers (it is $2\mapsto 3\mapsto 5\mapsto 2$ permutation in the detailed example). From the results obtained in the paper it follows in particular, that the ring $\mathbb{Z}$ is not an unique addition ring (UA-ring). Bibliography: 15 titles.

Détail

Keywords: ring of integers, $E$-ring, additive group, unique addition ring, multiplicative semigroup of a ring, monoid.