Abstract
We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of Handelman and a 2014 result of Garg, Grover, and Khurana, we prove that for any ring R, perspectivity is transitive in \(\mathbb {M}_{2}(R)\) if and only if R has stable range one. Also generalizing a 2019 result of Amini, Amini, and Momtahan we prove that a quasi-continuous module in which perspectivity is transitive is perspective.
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Amini, B., Amini, A., Momtahan, E.: Weakly perspective rings and modules. J. Algebra Appl. 18(1), 1950014, 9 pages (2019). MR 3910667
Diesl, A.J., Dittmer, S.J., Nielsen, P.P.: Idempotent lifting and ring extensions. J. Algebra Appl. 15(6), 1650112, 16 pages (2016). MR 3479816
Fuchs, L.: On a substitution property of modules. Monatsh. Math. 75, 198–204 (1971). MR 296096
Garg, S., Grover, H.K., Khurana, D.: Perspective rings. J. Algebra 415, 1–12 (2014). MR 3229504
Goodearl, K.R.: von Neumann regular rings, second ed. Krieger, R.E., Publishing Co., Inc, Malabar, FL (1991). MR 1150975
Handelman, D.: Perspectivity and cancellation in regular rings. J. Algebra 48(1), 1–16 (1977). MR 447329
Khurana, D., Lam, T.Y.: Rings with internal cancellation. J. Algebra 284(1), 203–235 (2005). MR 2115012
Khurana, D., Nielsen, P.P.: Perspectivity and von Neumann regularity, submitted, 1–17 (2020)
Lam, T.Y.: A crash course on stable range, cancellation, substitution and exchange. J. Algebra Appl. 3(3), 301–343 (2004). MR 2096452
Mohamed, S.H., Müller, B.J.: Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, vol. 147. Cambridge University Press, Cambridge (1990). MR 1084376
Vasershtein, L.N.: Stable rank of rings and dimensionality of topological spaces. Funktsional. Anal. i Prilozhen 5(2), 17–27 (1971). [English Translation: Funct. Anal. Appl. 5(2) 102–110 (1971).] MR 0284476
Warfield, R.B. Jr: Cancellation of modules and groups and stable range of endomorphism rings. Pacific J. Math. 91(2), 457–485 (1980). MR 615693
Acknowledgements
We thank Xavier Mary for comments on our paper, as well as sending us an alternative proof of Theorem 2.5.
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Presented by: Michel Brion
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Khurana, D., Nielsen, P.P. Transitivity of Perspectivity. Algebr Represent Theor 25, 281–287 (2022). https://doi.org/10.1007/s10468-020-10020-y
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DOI: https://doi.org/10.1007/s10468-020-10020-y
Keywords
- Dedekind-finite
- Quasi-continuous modules
- Stable range one
- (Transitivity of) perspectivity
- Von Neumann regular rings