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Injective modules over the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  22 June 2020

Gene Abrams ,
Francesca Mantese  and
Alberto Tonolo
Gene Abrams*
Affiliation:
Department of Mathematics, University of Colorado, 1420 Austin Bluffs Parkway, Colorado Springs, CO80918, USA
Francesca Mantese
Affiliation:
Dipartimento di Informatica, Università degli Studi di Verona, Strada le Grazie 15, 37134Verona, Italy e-mail: francesca.mantese@univr.it
Alberto Tonolo
Affiliation:
Dipartimento di Scienze Statistiche, Università degli Studi di Padova, via Cesare Battisti 241, 35121Padova, Italy e-mail: alberto.tonolo@unipd.it
*
e-mail: abrams@math.uccs.edu
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Abstract

For a field K, let $\mathcal {R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $ . We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal {R}$ -modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal {R}$ . Our approach involves realizing $\mathcal {R}$ up to isomorphism as the Leavitt path K-algebra of an appropriate graph $\mathcal {T}$ , which thereby allows us to utilize important machinery developed for that class of algebras.

Keywords

Jacobson algebrainjective moduleinjective envelopeLeavitt path algebraChen simple modulePrüfer module

MSC classification

Secondary: 16D50: Injective modules, self-injective rings
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 323 - 339
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Copyright
© Canadian Mathematical Society 2020

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References

Abrams, G., Ara, P., and Siles Molina, M., Leavitt path algebras . Lecture Notes in Mathematics, 2191, Springer-Verlag, London, 2017. https://doi.org/10.1007/978-1-4471-7344-1 Google Scholar
Abrams, G., Mantese, F., and Tonolo, A., Extensions of simple modules over Leavitt path algebras . J. Algebra 431(2015), 78–106. https://doi.org/10.1016/j.jalgebra.2015.01.034 CrossRefGoogle Scholar
Abrams, G., Mantese, F., and Tonolo, A., Leavitt path algebras are Bézout . Israel J. Math. 228(2018), 53–78. https://doi.org/10.1007/s11856-018-1773-2 CrossRefGoogle Scholar
Abrams, G., Mantese, F., and Tonolo, A., Prüfer modules over Leavitt path algebras . J. Algebra Appl. 18(2019), 1950154. https://doi.org/10.1142/s0219498819501548 CrossRefGoogle Scholar
Anderson, F. and Fuller, K., Rings and categories of modules . 2nd ed., Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992. https://doi.org/10.1007/978-1-4684-9913-1 Google Scholar
Ara, P. and Rangaswamy, K., Finitely presented simple modules over Leavitt path algebras . J. Algebra 417(2014), 333–352. https://doi.org/10.1016/j.jalgebra.2014.06.032 CrossRefGoogle Scholar
Bavula, V. V., The algebra of one-sided inverses of a polynomial algebra . J. Pure Appl. Algebra 214(2010), 1874–1897. https://doi.org/10.1016/j.jpaa.2009.12.033 CrossRefGoogle Scholar
Bergman, G., Coproducts and some universal ring constructions . Trans. Amer. Math. Soc. 200(1974), 33–88. https://doi.org/10.1090/s0002-9947-1974-0357503-7 CrossRefGoogle Scholar
Chen, X. W., Irreducible representations of Leavitt path algebras . Forum Math. 27(1)(2015), 549–574. https://doi.org/10.1515/forum-2012-0020 CrossRefGoogle Scholar
Cohn, P. M., Some remarks on the Invariant Basis property . Topology 5(1966), 215–228. https://doi.org/10.1016/0040-9383(66)90006-1 CrossRefGoogle Scholar
Gerritzen, L., Modules over the algebra of the noncommutative equation $\mathrm{yx}=1$ . Arch. Math. 75(2000), 98–112. https://doi.org/10.1007/pl00000437 CrossRefGoogle Scholar
Iovanov, M. and Sistko, A., On the Toeplitz-Jacobson algebra and direct finiteness . In: Groups, rings, group rings, and Hopf algebras, Contemporary Math, 668, Amer. Math. Soc., Providence, RI, 2017, pp. 113–124. https://doi.org/10.1090/conm/688/13830 CrossRefGoogle Scholar
Jacobson, N., Some remarks on one-sided inverses . Proc. Amer. Math. Soc. 1(1950), 352–355. https://doi.org/10.1007/978-1-4612-3694-8_6 CrossRefGoogle Scholar
Lam, T. Y., Lectures on modules and rings . Graduate Texts in Mathematics, 189, Springer-Verlag, Berlin Heidelberg, 1999. https://doi.org/10.1007/978-1-4612-0525-8 Google Scholar
Lu, Z., Wang, L., and Wang, X., Nonsplit module extensions over the one-sided inverse of $k\left[x\right]$ . Involve 12(8)(2019), 1369–1377. https://doi.org/10.2140/involve.2019.12.1369 CrossRefGoogle Scholar
Rangaswamy, K. M., On simple modules over Leavitt path algebras . J. Algebra 423(2015), 239–258. https://doi.org/10.1016/j.jalgebra.2014.10.008 CrossRefGoogle Scholar