Finite completely primary rings in which the product of any two zero divisors of a ring is in its coefficient subring (Q1335493)

The paper deals with finite completely primary rings. On any Galois ring \(S\) of the form \(\text{GR}(p^ n,n)\) a construction of a new ring is given. It is proved that in any completely finite primary ring the product of any two zero-divisors is an element of its coefficient subring if and only if it is one of the rings given by the defined construction. It is proved that finite rings in which the product of any two zero divisors is a power of a fixed prime \(q\) are completely primary rings where either the set of zero divisors \(J\) has \(J^ 2 = 0\) or their coefficient subring is \(Z_{2^ n}\) with \(n = 2\) or 3. Finally, the enumeration of such rings is determined.