Finite Local Rings Free of Rank Three over Zpn: Radical Filtrations, Matrix Models, and the Frobenius Property
- 1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Abstract
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.
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BibTeX
@article{saif2026finite,
title = {{Finite Local Rings Free of Rank Three over Zpn: Radical Filtrations, Matrix Models, and the Frobenius Property}},
author = {Sami H. Saif and Yousef Alkhamees},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100755},
url = {https://www.mdpi.com/2075-1680/15/10/755},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Finite Local Rings Free of Rank Three over Zpn: Radical Filtrations, Matrix Models, and the Frobenius Property
AU - Sami H. Saif
AU - Yousef Alkhamees
PY - 2026
DA - 2026/10/10
JO - Axioms
DO - 10.3390/axioms15100755
UR - https://www.mdpi.com/2075-1680/15/10/755
AB - 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.
SN - 2075-1680
ER -