  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].

Rings and Algebras