Abstract
In this paper the author constructs a chain ring (i.e. a ring in which the right and left ideals are linearly ordered by inclusion) with the following properties: 1) is a prime ring; 2) the Jacobson radical of is a simple chain ring (without identity); 3) each element of is a right and left zero divisor. This example gives an answer to one of Brung's questions. In addition, the ring 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.
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