Sheiham, Desmond [Ranicki, Andrew; Schofield, Aidan] Universal localization of triangular matrix rings. (English) Zbl 1115.16010 Proc. Am. Math. Soc. 134, No. 12, 3465-3474 (2006). 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 \(N\). Reviewer: Radoslav M. Dimitrić (Uniontown) Cited in 1 ReviewCited in 6 Documents 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) Keywords:universal localizations; \(\sigma\)-inverting morphisms; localizations as pushouts; finitely generated projective modules; rings of fractions; triangular matrix rings × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] P. Ara, M. A. González-Barroso, K. R. Goodearl, and E. Pardo, Fractional skew monoid rings, J. Algebra 278 (2004), no. 1, 104 – 126. · Zbl 1063.16033 · doi:10.1016/j.jalgebra.2004.03.009 [2] D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. · Zbl 0718.20001 [3] George M. Bergman, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1 – 32. · Zbl 0264.16017 [4] George M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc. 200 (1974), 33 – 88. · Zbl 0264.16018 [5] George M. Bergman and Warren Dicks, Universal derivations and universal ring constructions, Pacific J. Math. 79 (1978), no. 2, 293 – 337. · Zbl 0359.16001 [6] P. M. Cohn. Localization in general rings, a historical survey. Proceedings of the Conference on Noncommutative Localization in Algebra and Topology, ICMS, Edinburgh, 29-30 April, 2002, London Mathematical Society Lecture Notes 330, Cambridge University Press, 5-23, 2006. · Zbl 1125.16019 [7] P. M. Cohn, Free rings and their relations, Academic Press, London-New York, 1971. London Mathematical Society Monographs, No. 2. · Zbl 0232.16003 [8] P. M. Cohn, Rings of fractions, Amer. Math. Monthly 78 (1971), 596 – 615. · Zbl 0212.37501 · doi:10.2307/2316568 [9] P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. · Zbl 0659.16001 [10] P. M. Cohn and Warren Dicks, Localization in semifirs. II, J. London Math. Soc. (2) 13 (1976), no. 3, 411 – 418. · Zbl 0328.16003 · doi:10.1112/jlms/s2-13.3.411 [11] Warren Dicks and Eduardo D. Sontag, Sylvester domains, J. Pure Appl. Algebra 13 (1978), no. 3, 243 – 275. · Zbl 0393.16002 · doi:10.1016/0022-4049(78)90011-7 [12] M. Farber and P. Vogel, The Cohn localization of the free group ring, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 433 – 443. · Zbl 0767.16006 · doi:10.1017/S0305004100075538 [13] A. Haghany and K. Varadarajan, Study of formal triangular matrix rings, Comm. Algebra 27 (1999), no. 11, 5507 – 5525. · Zbl 0941.16005 · doi:10.1080/00927879908826770 [14] A. Haghany and K. Varadarajan, Study of modules over formal triangular matrix rings, J. Pure Appl. Algebra 147 (2000), no. 1, 41 – 58. · Zbl 0951.16009 · doi:10.1016/S0022-4049(98)00129-7 [15] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. · Zbl 0911.16001 [16] A. A. Ranicki. Noncommutative localization in topology. Proceedings of the Conference on Noncommutative Localization in Algebra and Topology, ICMS, Edinburgh, 29-30 April, 2002. arXiv:math.AT/0303046, London Mathematical Society Lecture Notes 330, Cambridge University Press, 81-102, 2006. · Zbl 1125.55004 [17] A. H. Schofield, Representation of rings over skew fields, London Mathematical Society Lecture Note Series, vol. 92, Cambridge University Press, Cambridge, 1985. · Zbl 0571.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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