Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T04:07:51.824Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Alberto Tonolo
Affiliation:
Dipartimento di Scienze Statistiche, Università degli Studi di Padova, via Cesare Battisti 241, 35121Padova, Italy e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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