Abstract
We raise the following general question regarding a ring graded by a group: “If P is a ring-theoretic property, how does one define the graded version Pgr of the property P in a meaningful way?”. Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If P is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first part of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness. In the second part of the paper, we consider graded matrix and Leavitt path algebras. If K is a trivially graded field and E is a directed graph, the Leavitt path algebra LK(E) is naturally graded by the ring of integers. If E is a finite graph, we present a property of E which is equivalent with LK(E) being graded unit-regular. This property critically depends on the lengths of paths to cycles and it further illustrates that graded unit-regularity is quite restrictive in comparison to the alternative generalization of unit-regularity from the first part of the paper.
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Abrams, G., Ara, P., Siles Molina, M.: Leavitt path algebras, Lecture Notes in Mathematics 2191, p 2017. Springer, London (2191)
Abrams, G., Rangaswamy, K.M.: Regularity conditions for arbitrary Leavitt path algebras. Algebr. Represent. Theory 13(3), 319–334 (2010)
Aranda Pino, G., Rangaswamy, K.M., Vaš, L.: ∗-regular Leavitt path algebra of arbitrary graphs. Acta Math. Sci. Ser. BEngl. Ed. 28(5), 957–968 (2012)
Ara, P., Hazrat, R., Li, H., Sims, A.: Graded Steinberg algebras and their representations. Algebra Number Theory 12(1), 131–172 (2018)
Berberian, S.K.: Baer rings and Baer ∗-rings, 1988, preprint at https://www.ma.utexas.edu/mp_arc/c/03/03-181.pdf
Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger Publishing Co., Malabar (1991)
Hazrat, R.: Leavitt path algebras are graded von Neumann regular rings. J. Algebra 401, 220–233 (2014)
Hazrat, R.: Graded rings and graded Grothendieck groups, London Math. Soc. Lecture Note Ser. 435, Cambridge Univ. Press (2016)
Hazrat, R., Rangaswamy, K.M., Srivastava, A.K.: Leavitt path algebras: graded direct-finiteness and graded \(\sum \)-injective simple modules. J. Algebra 503, 299–328 (2018)
Hazrat, R., Vaš, L.: Baer and Baer ∗-ring characterizations of Leavitt path algebras. J. PureAppl. Algebra 222(1), 39–60 (2018)
Hazrat, R., Vaš, L.: K-theory classification of graded ultramatricial algebras with involution. Forum Math. 31(2), 419–463 (2019)
Lam, T.Y., Murray, W.: Unit regular elements in corner rings. Bull. Hong Kong Math. Soc. 1(1), 61–65 (1997)
Lam, T.Y.: A crash course on stable range, cancellation, substitution and exchange. J. Algebra Appl. 03, 301–343 (2004)
Năstăsescu, C., van Oystaeyen, F.: Methods of Graded Rings, Lecture Notes in Mathematics, p 2004. Springer, Berlin (1836)
Vaš, L.: Canonical traces and directly finite Leavitt path algebras. Algebr. Represent. Theory 18, 711–738 (2015)
Vaš, L.: Simplicial and dimension groups with group action and their realization, submitted for publication, preprint arXiv:1805.07636 [math.KT]
Vaš, L.: Realization of graded matrix algebras as Leavitt path algebras, Beitr. Algebra Geom., in print, preprint arXiv:1910.05174 [math.RA]
Vaserstein, L.N.: Stable rank of rings and dimensionality of topological spaces. Funktsional. Anal. i Prilozhen. 5, 17–27 (1971). English translation:, Funct. Anal. Appl., 5 (1971), 102–110
Vaserstein, L.N.: Bass’s first stable range condition. J. Pure Appl. Algebra 34, 319–330 (1984)
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Presented by: Kenneth Goodearl
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A part of this paper was conceived during the Asia-Australia Algebra Conference and the author’s visit to Western Sydney University in January 2019. This visit was supported by the NSF-AWM Travel Grants program.
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Vaš, L. Graded Cancellation Properties of Graded Rings and Graded Unit-regular Leavitt Path Algebras. Algebr Represent Theor 24, 625–649 (2021). https://doi.org/10.1007/s10468-020-09963-z
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DOI: https://doi.org/10.1007/s10468-020-09963-z
Keywords
- Graded ring
- Unit-regular ring
- Cancellability
- Stable range 1
- Directly finite ring
- Graded matrix algebra
- Leavitt path algebra