ON INFINITE MACWILLIAMS RINGS AND MINIMAL INJECTIVITY CONDITIONS

We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560-576] and Schneider and Zumbr<spacing diaeresis>agel [Proc. Amer. Math. Soc. 147 (2019), pp. 947-961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947-961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general "minimal test for injectivity" for a left Artinian ring A: we show that if every injective morphism Sigma(k) -> A from the k'th socle of A extends to a morphism A -> A, then the ring is quasi-Frobenius. We also give a general result under which if injective maps N -> M from submodules N of a module M extend to endomorphisms of M (pseudo-injectivity), then all such morphisms N -> M extend (quasi-injectivity) and obtain further applications.