The MacWilliams extending property and pseudo-Frobenius rings with essential socles

Quynh T. C., KOŞAN M. T.

Journal of Algebra and its Applications, 2025 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1142/s0219498826501860
  • Dergi Adı: Journal of Algebra and its Applications
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Anahtar Kelimeler: automorphism-invariant module and ring, Hamming weight preserving homomorphism, linear code, MacWilliams module and ring, monomial transformation, perfect rings, pseudo-Frobenius ring, quasi-Frobenius ring, self-injective ring
  • Gazi Üniversitesi Adresli: Evet

Özet

In this paper, we study n-MacWilliams right R-modules. It is shown that (1) the MacWilliams extension property for n = 1 implies that MR is an automorphism-invariant module (the converse holds if MR is finitely cogenerated), (2) for n = 2, the MacWilliams extension property implies that MR is quasi-injective which yields that if the matrix ring Mn(R) over a right finitely cogenerated ring R is right 1-MacWilliams, then R is also right 1-MacWilliams. We also show that (3) a right automorphism-invariant ring containing no infinite orthogonal sets of idempotents with essential right socle is right pseudo-Frobenius and (4) if R is a right 1-MacWilliams ring and it has ACC on right annihilators, then R is quasi-Frobenius which generalize answers the question of when is a MacWilliams ring quasi-Frobenius and when is a quasi-Frobenius ring a right artinian, right automorphism-invariant?