An example of a chain prime ring with nilpotent elements

In this paper the author constructs a chain ring $R$ (i.e. a ring in which the right and left ideals are linearly ordered by inclusion) with the following properties: 1) $R$ is a prime ring; 2) the Jacobson radical $J(R)$ of $R$ is a simple chain ring (without identity); 3) each element of $J(R)$ is a right and left zero divisor. This example gives an answer to one of Brung's questions. In addition, the ring $J(R)$ is totally singular, i.e. it coincides with its right (left) singular ideal.
The construction is based on a theorem that permits one to assign a chain ring to a right ordered group whose group ring can be imbedded in a division ring.
Bibliography: 9 titles.