Frobenius–Artin algebras and infinite linear codes

We generalize the results on finite Frobenius rings of T. Honold (2001) [16] and some classical results of Nakayama (1939, 1941) [21] and [22] on Frobenius algebras over fields, and the results of J.A. Wood (2008, 1999) [31] and [32] on linear codes and finite Frobenius rings, to the setting of Artin algebras, and provide a unifying context for these results. We show that an Artin algebra is Frobenius if and only if its socle and top are isomorphic only as left modules (equivalently, as right modules). We show that an Artin algebra A satisfies the MacWilliams code equivalence property if and only if A is a product of a finite Frobenius ring and a quasi-Frobenius ring with no nontrivial finite representations. We use a blend of ring theoretic, combinatorial and compact group methods; in particular, inspired by the work of J.A. Wood, we show how the theory of compact groups can be used to yield ring theoretical results.