Completability of Unimodular Rows in Pure Projective Modules: A Local-Global Perspective

In this paper, we extend classical results on unimodular elements and the completability of unimodular rows in torsion-free pure projective modules over commutative Noetherian rings. We establish a stable freeness criterion that characterises global completability in terms of module-theoretic properties, generalising Serre-Plumstead-type results for a broader class of modules that are not necessarily free. Applications to linear codes over rings are investigated, showing strong structural and distance properties. This work generalises the local-global principle for row completability beyond projective modules and provides a new framework for understanding module decompositions in the algebraic case. Applications include the construction of generator matrices in coding theory over rings, where unimodular rows encode critical structural and distance properties.