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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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: Let $ R$ be an algebra over a field $ k$, and $ P,Q$ be two nonzero finitely generated projective $ R$-modules. By adjoining further generators and relations to $ R$, one can obtain an extension $ S$ of $ R$ having a universal isomorphism of modules, $ i:P{ \otimes _R}S \cong Q{ \otimes _R}S$.

We here study this and several similar constuctions, including (given a single finitely generated projective $ R$-module $ P$) the extension $ S$ of $ R$ having a universal idempotent module-endomorphism $ e:P \otimes S \to P \otimes S$, and (given a positive integer $ n$) the $ k$-algebra $ S$ with a universal $ k$-algebra homomorphism of $ R$ into its $ n \times n$ matrix ring, $ f:R \to {\mathfrak{m}_n}(S)$.

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 $ {R_0}$ ( $ ( = k \times k \times k,k \times k$, and $ k$ 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 $ S$ is the same as that of $ R$ 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 $ [P] = [Q]$.

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 $ R$ of a universal inverse to a given homomorphism of finitely generated projective modules, $ f:P \to Q$, and the formation of the factor-ring $ R/{T_P}$ by the trace ideal of a given finitely generated projective $ R$-module $ P$ (in other words, setting $ P = 0$).

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

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Keywords: Universal module homomorphism (isomorphism), finitely generated projective module, matrix ring, coproduct of rings, representable functor, semigroup of $ R$-modules (resp. projective $ R$-modules) under ``$ \oplus $", global dimension, $ n$-fir, localization, trace ideal of a projective module, $ k$-linear category
Article copyright: © Copyright 1974 American Mathematical Society