Finite Local Rings Free of Rank Three over Zpn: Radical Filtrations, Matrix Models, and the Frobenius Property

Let p be a prime, let n≥2, and let R be a finite commutative local ring that is free of rank three over R0=Zpn and has residue field Fp. We show that R/pR is isomorphic either to Fp[t]/(t3) or to Fp[s,t]/(s,t)2, and that this dichotomy governs the ideal theory of R. In the first case, R≅R0[X]/(f) for a monic cubic polynomial f≡X3(modp); the ring R is a chain ring if p2∤f(0) and has embedding dimension two if p2∣f(0). In embedding dimension two, the nilpotency index t of the Jacobson radical J satisfies the sharp bounds n+2≤t≤2n+1, and we characterize the rings with J3=pJ2, for which all powers of J are given explicitly. In the second case, the embedding dimension is three, Ji=pi−1J for all i≥1, and t=n+1. In particular, no ring of embedding dimension two satisfies p∈J2. Using the left regular representation, we derive and solve the associativity equations for the structure constants and obtain an explicit six-parameter matrix model, together with matrix criteria for locality and for isomorphism. Finally, we show that R is a Frobenius ring if and only if its embedding dimension is at most two.