A classification of module braces over the ring of $\bold p$-adic integers

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 integers and $\cdot$ is the product of the commutative and $3$-nilpotent $R$-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 algebras are in correspondence with particular equivalence classes of the bilinear forms defined starting from the products of the algebras.