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Injective modules over the Jacobson algebra
$K\langle X, Y \ | \ XY=1\rangle $
Published online by Cambridge University Press: 22 June 2020
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.
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- © Canadian Mathematical Society 2020
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