MacWilliams extending conditions and quasi-Frobenius rings

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 [13], [22]. 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 [13].