Constructing division rings as module-theoretic direct limits

Abstract

If R R is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring D D generated by a homomorphic image of R R is a rule putting on all free R R -modules of finite rank matroid structures (closure operators satisfying the exchange axiom) subject to certain functoriality conditions. This note gives a new description of how D D may be constructed from this data. (A classical precursor of this is the construction of Q \mathbf Q as a field with additive group a direct limit of copies of Z \mathbf Z .) The division rings of fractions of right and left Ore rings, the universal division ring of a free ideal ring, and the concept of a specialization of division rings are then interpreted in terms of this construction.