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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, CO 80918, USA
Francesca Mantese
Affiliation:
Dipartimento di Informatica, Università degli Studi di Verona, Strada le Grazie 15, 37134 Verona, Italy e-mail: francesca.mantese@univr.it
Alberto Tonolo
Affiliation:
Dipartimento di Scienze Statistiche, Università degli Studi di Padova, via Cesare Battisti 241, 35121 Padova, 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

Information

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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