Semiperfect rings with a Nakayama permutation: A survey of Double annihilator property and Size condition

Abstract

For a semiperfect ring with essential socles, the Double annihilator property encodes that the top and socle have anti-isomorphic lattices of submodules, whereas the Size condition encodes that they are isomorphic as modules. Interest in both concepts, particularly for finite rings, was revived by coding theory, where they characterise QF rings and Frobenius rings, respectively. However, their shared origins date back to the work of T. Nakayama. We study these concepts through the lens of the Nakayama permutation, an invariant initially used to define (quasi-)Frobenius rings. We propose semiperfect rings as the setting for this study, treating them as the natural generalisation of finite rings, because they possess the characteristic decomposition of unity preserved by projection onto a semisimple top. This allows us to extend the utility of the Nakayama permutation beyond the classical Artinian setting. By analysing the Nakayama permutation in this broader context, we show that many classical properties of (quasi-)Frobenius rings are not exclusive to the finite case, but are special cases of the general behaviour of semiperfect rings with essential socles. We illustrate these results using B. J. M\"uller's representation of semiperfect rings as rings of formal matrices. The clear description of socles and tops in this setting provides a straightforward method for constructing counterexamples, such as quasi-Frobenius rings that are not Frobenius.