MacWilliams' extension theorem for infinite rings

TL;DR

This paper generalizes MacWilliams' extension theorem from finite Frobenius rings to left Artinian rings, characterizing when these rings have the MacWilliams property through algebraic and topological conditions.

Contribution

It extends the MacWilliams extension property characterization to Artinian rings, linking algebraic properties with topological perspectives.

Findings

A left Artinian ring has the left MacWilliams property iff it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient.

The finitary left socle embeds into the semisimple quotient iff it admits a finitarily left torsion-free character.

An Artinian ring has the MacWilliams property iff it is finitarily Frobenius, i.e., quasi-Frobenius with socle embedding conditions.

Abstract

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.