Leavitt path algebras: Graded direct-finiteness and graded Σ-injective simple modules

JOURNAL OF ALGEBRA

卷:503

Article

Hazrat, Roozbeh¹  Rangaswamy, Kulumani M.²  Srivastava, Ashish K.³ 

[1] Western Sydney Univ, Ctr Res Math, Penrith, NSW, Australia

[2] Univ Colorado, Dept Math, Colorado Springs, CO 80918 USA

[3] St Louis Univ, Dept Math & Stat, St Louis, MO 63103 USA

关键词: Leavitt path algebras;    Bounded index of nilpotence;    Direct-finiteness;    Simple modules;    Injective modules;    Sigma-injective modules;   

DOI  :  10.1016/j.jalgebra.2018.01.041

来源: Elsevier

【 摘 要 】

In this paper, we give a complete characterization of Leavitt path algebras which are graded Sigma-V rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra L over an arbitrary graph E is a graded E-V ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over K or K[x, x(-1)] with appropriate matrix gradings. We also obtain a graphical characterization of such a graded Sigma-V ring L. When the graph E is finite, we show that L is a graded Sigma-V ring double left right arrow L is graded directly-finite double left right arrow L has bounded index of nilpotence double left right arrow L is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph B is infinite. Following this, we also characterize Leavitt path algebras L which are non-graded Sigma-V rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra L is a graded Sigma-V ring, then L is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Vas [33] on directly-finite Leavitt path algebras. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free