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Frobenius Local Rings of Length 5 and Index of Nilpotency 3

Author

Listed:
  • Sami Alabiad

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Alhanouf Ali Alhomaidhi

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

Abstract

This paper investigates finite local non-chain rings associated by the well-known invariants p , n , m , l , and k , where p is a prime number. In particular, we provide a comprehensive characterization of Frobenius local rings of length l = 5 and index of nilpotency t = 3 , where t the index of nilpotency of the maximal ideal. The relevance of Frobenius rings is notable in coding theory, as it has been demonstrated that two classical results by MacWilliams—the Extension Theorem and the MacWilliams identities—are applicable not only to finite fields but also to finite Frobenius rings. We, therefore, classify and count Frobenius local rings of order p 5 m with t = 3 , outlining their properties in connection with various values of n .

Suggested Citation

  • Sami Alabiad & Alhanouf Ali Alhomaidhi, 2025. "Frobenius Local Rings of Length 5 and Index of Nilpotency 3," Mathematics, MDPI, vol. 13(5), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:781-:d:1600720
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    References listed on IDEAS

    as
    1. Sami Alabiad & Alhanouf Ali Alhomaidhi & Nawal A. Alsarori, 2024. "On Linear Codes over Local Rings of Order p 4," Mathematics, MDPI, vol. 12(19), pages 1-20, September.
    2. Sami Alabiad & Alhanouf Ali Alhomaidhi & Nawal A. Alsarori, 2024. "On Linear Codes over Finite Singleton Local Rings," Mathematics, MDPI, vol. 12(7), pages 1-14, April.
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