On the structure of one-generated Leibniz rings

Abstract

The paper is devoted to the study of one-generated Leibniz rings. We begin with a short discussion of the relation between Lie rings, Lie algebras, Leibniz algebras, and Leibniz rings, and recall the basic notions needed in the sequel. A special construction of Leibniz rings generated by one element is introduced and investigated. It is shown that these rings play the role of free objects in the class of one-generated Leibniz rings. In particular, every one-generated Leibniz ring is an epimorphic image of the Leibniz ring $$$FL(1,\mathbb{Z})$$$, and the finite-exponent case is treated by means of the corresponding rings $$$FL(1,\mathbb{Z}_{k})$$$. We also present an equivalent description of $$$FL(1,\mathbb{Z})$$$ in terms of the polynomial ring $$$\mathbb{Z}[X]$$$. Further, the structure of one-generated Leibniz rings is considered in several important cases, including torsion-free additive groups of finite $$$0$$$-rank and additive groups which are elementary abelian $$$p$$$-groups.