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Published online by Cambridge University Press: 05 November 2011
One of the main results of Ch. 4 stated that every semifir has a universal field of fractions. This is now applied to show that every family of fields all having a common subfield can be embedded in a universal fashion in a field, their field coproduct. We begin in 5.1 by explaining the coproduct construction for groups (where it is relatively simple) and for rings, and derive some of the simpler consequences when the common subring is a field. When the factors themselves are fields, an elaboration of these results will show the ring coproduct of fields to be a fir (by an analogue of the weak algorithm, see Cohn [60, 61]), but we shall not follow this route, since it will appear as a consequence of more general later results.
The study of coproducts requires a good deal of notation; some of this is introduced in 5.2 and is used there to define the module induced by a family of modules over the factor rings and compute its homological dimension. In 5.3 we prove the important coproduct theorems of Bergman [74]: If R is the ring coproduct of a family (Rλ) of rings, taken over a field K, then (i) the global dimension of R is the supremum of the global dimensions of the factors (or possibly 1 if all the factors have global dimension 0) (Th. 3.5), (ii) the monoid of projectives P(R) is the coproduct of the P(Rλ) over P(K) (Th. 3.8).
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