Strongly stable rank and applications to matrix completion

ABSTRACT

We perform an in-depth study of strongly stable ranks of modules over a commutative ring. Here we define the strongly stable rank of a module to be the supremum of the stable ranks of its finitely generated submodules. As an application, we give non-Noetherian generalizations of known facts about outer products and matrix completions over PIRs and Dedekind domains. We construct Noetherian and non-Noetherian domains of arbitrary strongly stable rank. We also consider strongly n-generated ideals, and we characterize the rings in which every ideal is strongly 2-generated and the domains in which every ideal is strongly 3-generated.

KEYWORDS:

• Completable matrix
• outer product
• stable rank
• strongly n-generated ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F05
• 13F10
• 13F30
• 15A83
• 19B10