A classification of module braces over the ring of $\mathbf{p}$-adic integers

TL;DR

This paper classifies certain algebraic structures called module braces over the ring of p-adic integers, providing a detailed classification in both torsion-free and torsion cases based on bilinear forms.

Contribution

It introduces a classification scheme for R-braces over p-adic integers, linking algebraic structures to bilinear forms and distinguishing cases by isomorphism and isoclinism.

Findings

Classification up to isomorphism for torsion-free cases

Classification up to isoclinism for torsion cases

Correspondence between algebra classes and bilinear form equivalence classes

Abstract

In this paper we study the $R$-braces $(M,+,\circ)$ such that $M\cdot M$ is cyclic, where $R$ is the ring of $p$-adic and $\cdot$ is the product of the radical $R$-algebra associated to $M$. In particular, we give a classification up to isomorphism in the torsion-free case and up to isoclinism in the torsion case. More precisely, the isomorphism classes and the isoclinism classes of such radical algebras are in correspondence with particular equivalence classes of the bilinear forms defined starting from the products of the algebras.