## Universal localization of triangular matrix rings.(English)Zbl 1115.16010

This is a posthumous paper that was prepared for publication by two of the mathematicians whose results the author generalizes. Assume that $$R$$ is an associative unital ring and let $$\sigma\colon P\to Q$$ denote a morphism between finitely generated projective $$R$$-modules. There is a ring morphism (called a ‘universal localization’) $$R\to\sigma^{-1}R$$ which is universal with respect to the property that $$\sigma^{-1}R\otimes_RP\subseteq 1\otimes\sigma\sigma^{-1}R\otimes_RQ$$ is an isomorphism (as established by Cohn, Bergman, Schofield). By Bergman, the ring of fractions is obtained in case $$R$$ is commutative. If $$R$$ is non-commutative, understanding universal localizations is harder but a number of examples with elegant description of $$\sigma^{-1}R$$ exist.
This paper describes and generalizes some interesting examples due to A. Schofield with applications in topology. If $$R$$ denotes the triangular ring $$\left(\begin{smallmatrix} A&M\\ 0&B\end{smallmatrix}\right)$$ where $$A$$ and $$B$$ are associative unital rings and $$M$$ is an $$(A,B)$$-bimodule, then the columns are finitely generated projective left $$R$$-modules whose direct sum is isomorphic to $$R$$. The main result then states the following: Let $$\sigma$$ be a morphism of the first column module to the second column module of the said triangular ring matrix and let, for some $$p$$ we have $$\sigma((1\;0)^t)=(p\;0)^t$$; denoted $$T=T(M,p)$$. Then the universal localization is isomorphic to: $\left(\begin{smallmatrix} A&M\\ 0&B\end{smallmatrix}\right)\subseteq\left(\begin{smallmatrix}\rho_A &\rho_M\\ 0&\rho_B\end{smallmatrix}\right)\left(\begin{smallmatrix} T&T\\ T&T\end{smallmatrix}\right).$ Among other results, the author describes universal localization $$\sigma^{-1}R\otimes_RN$$ of an $$R$$-module $$N$$.

### MSC:

 16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 16S50 Endomorphism rings; matrix rings 16E20 Grothendieck groups, $$K$$-theory, etc. 16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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