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

According to general terminology, a ringRis completely primary if its set of zero divisorsJforms an ideal. LetRbe a finite completely primary ring. It is easy to establish thatJis the unique maximal ideal ofRandRhas a coefficient subringS(i.e.R/Jisomorphic toS/pS) which is a Galois ring. In this paper we give the construction of finite completely primary rings in which the product of any two zero divisors is inSand determine their enumeration. We also show that finite rings in which the product of any two zero divisors is a power of a fixed prime p are completely primary rings with eitherJ2=0or their coefficient subring isZ2nwithn=2or3. A special case of these rings is the class of finite rings, studied in [2], in which the product of any two zero divisors is zero.