Constructing division rings as module-theoretic direct limits

Bergman, George

doi:10.1090/S0002-9947-02-02927-6

Constructing division rings as module-theoretic direct limits
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Trans. Amer. Math. Soc. 354 (2002), 2079-2114 Request permission

Abstract:

If $R$ is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring $D$ generated by a homomorphic image of $R$ is a rule putting on all free $R$-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 $D$ may be constructed from this data. (A classical precursor of this is the construction of $\mathbf Q$ as a field with additive group a direct limit of copies of $\mathbf Z$.) 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.

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