Injective modules over the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $
Canadian mathematical bulletin, Tome 64 (2021) no. 2, pp. 323-339
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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.
Mots-clés :
Jacobson algebra, injective module, injective envelope, Leavitt path algebra, Chen simple module, Prüfer module
Abrams, Gene; Mantese, Francesca; Tonolo, Alberto. Injective modules over the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $. Canadian mathematical bulletin, Tome 64 (2021) no. 2, pp. 323-339. doi: 10.4153/S0008439520000478
@article{10_4153_S0008439520000478,
author = {Abrams, Gene and Mantese, Francesca and Tonolo, Alberto},
title = {Injective modules over the {Jacobson} algebra $K\langle X, Y \ | \ XY=1\rangle $},
journal = {Canadian mathematical bulletin},
pages = {323--339},
year = {2021},
volume = {64},
number = {2},
doi = {10.4153/S0008439520000478},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000478/}
}
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%0 Journal Article %A Abrams, Gene %A Mantese, Francesca %A Tonolo, Alberto %T Injective modules over the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $ %J Canadian mathematical bulletin %D 2021 %P 323-339 %V 64 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000478/ %R 10.4153/S0008439520000478 %F 10_4153_S0008439520000478
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