Injective modules over the Jacobson algebra
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Published:2020-06-22
Issue:2
Volume:64
Page:323-339
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ISSN:0008-4395
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Container-title:Canadian Mathematical Bulletin
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language:en
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Short-container-title:Can. Math. Bull.
Author:
Abrams Gene,Mantese Francesca,Tonolo Alberto
Abstract
AbstractFor 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.
Publisher
Canadian Mathematical Society
Subject
General Mathematics