On infinite MacWilliams rings and minimal injectivity conditions

Abstract

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 A : we show that if every injective morphism Σ k → A \Sigma _k\rightarrow A from the k k ’th socle of A A extends to a morphism A → A A\rightarrow A , then the ring is quasi-Frobenius. We also give a general result under which if injective maps N → M N\rightarrow M from submodules N N of a module M M extend to endomorphisms of M M (pseudo-injectivity), then all such morphisms N → M N\rightarrow M extend (quasi-injectivity) and obtain further applications.