On Symmetrized Weight Compositions

Abstract

A characterization of module alphabets with the Hamming weight EP (abbreviation for Extension Property) had been settled. A thoughtfully constructed piece-of-art example by J.A.Wood ([7]) finished the tour. In 2009, in [8], Frobenius bimodules were proved to satisfy the EP with respect to symmetrized weight compositions. In [4], the embeddability in the character group of the ambient ring R was found sufficient for a module R A to satisfy the EP with respect to swc built on any subgroup of Aut R (A), while the necessity remained a question. Here, landing in a "Midway", the necessity is proved by jumping to Hamming weight. Corollary 1.11 declares a characterization of module alphabets satisfying the EP with respect to swc.Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for functions is that inputs are to the left.