Enumeration of Frobenius Local Rings of Order p6r via Bilinear Forms

Let p be a prime number and r a positive integer. This paper investigates the construction and classification of finite commutative rings of order p6r in which the set of zero-divisors J forms an ideal satisfying the conditions J3=0, J2≠0 with J2 being principal. Under these conditions, the rings considered are precisely the Frobenius local rings. A Frobenius local (completely primary) ring R with these properties is referred to as a ring with property (P). These rings naturally divide into three classes according to their characteristic: p, p2, or p3. In the case of characteristic p2, a further distinction is made depending on whether p lies in J2 or in J∖J2, where J denotes the Jacobson radical of R. The classification is achieved by associating to each ring a canonical matrix corresponding to a bilinear form and then applying matrix congruence techniques to reduce the problem to linear algebra over finite fields. This yields a complete and explicit description of all Frobenius local rings with property (P) of order p6r, including their algebraic structure and enumeration.