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 Zumbrägel [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\rightarrow A from the k'th socle of A extends to a morphism A\rightarrow A, then the ring is quasi-Frobenius. We also give a general result under which if injective maps N\rightarrow M from submodules N of a module M extend to endomorphisms of M (pseudo-injectivity), then all such morphisms N\rightarrow M extend (quasi-injectivity) and obtain further applications. [ABSTRACT FROM AUTHOR]