Coproducts and some universal ring constructions

Author:
George M. Bergman

Journal:
Trans. Amer. Math. Soc. **200** (1974), 33-88

MSC:
Primary 16A64

MathSciNet review:
0357503

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebra over a field , and be two nonzero finitely generated projective -modules. By adjoining further generators and relations to , one can obtain an extension of having a *universal* isomorphism of modules, .

We here study this and several similar constuctions, including (given a single finitely generated projective -module ) the extension of having a universal idempotent module-endomorphism , and (given a positive integer ) the -algebra with a universal -algebra homomorphism of into its matrix ring, .

A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring ( , and respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of is the same as that of unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation .

These results allow one to construct a large number of unusual examples.

We discuss the problem of obtaining similar results for some related constructions: the adjunction to of a universal inverse to a given homomorphism of finitely generated projective modules, , and the formation of the factor-ring by the trace ideal of a given finitely generated projective -module (in other words, setting ).

The idea for a category-theoretic generalization of the ideas of the paper is also sketched.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1974-0357503-7

Keywords:
Universal module homomorphism (isomorphism),
finitely generated projective module,
matrix ring,
coproduct of rings,
representable functor,
semigroup of -modules (resp. projective -modules) under ``",
global dimension,
-fir,
localization,
trace ideal of a projective module,
-linear category

Article copyright:
© Copyright 1974
American Mathematical Society