MacWilliams extending conditions and quasi-Frobenius rings

TL;DR

This paper investigates conditions under which MacWilliams rings are quasi-Frobenius, establishing new links between ring properties like noetherian, perfect, and automorphism-invariant, and their classification as quasi-Frobenius.

Contribution

It proves that right or left noetherian left 1-MacWilliams rings are quasi-Frobenius, and that right perfect, left automorphism-invariant rings are left self-injective, answering open questions.

Findings

Noetherian left 1-MacWilliams rings are quasi-Frobenius.

Right perfect, left automorphism-invariant rings are left self-injective.

Artinian, left automorphism-invariant rings are quasi-Frobenius.

Abstract

MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/proc/15929] and [F. M. Schneider, J. Zumbr\"{a}gel, MacWilliams' extension theorem for infinite rings, Proc. Amer. Math. Soc. 147, 3 (2019), 947-961]. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that if $R$ is a right (or left) artinian, left automorphism-invariant ring, then $R$ is quasi-Frobenius, thus answering a question asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/proc/15929].