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

Roozbeh Hazrat, Kulumani M. Rangaswamy, Ashish K. Srivastava

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

In this paper, we give a complete characterization of Leavitt path algebras which are graded Σ-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 Lover an arbitrary graph E is a graded Σ-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 Σ-V ring L. When the graph E is finite, we show that Lis a graded Σ-V ring ⇐⇒Lis graded directly-finite ⇐⇒L has bounded index of nilpotence ⇐⇒Lis graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph E is infinite. Following this, we also characterize Leavitt path algebras L which are non-graded Σ-V rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra Lis a graded Σ-V ring, then Lis 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 Vaš [33] on directly-finite Leavitt path algebras.
Original languageEnglish
Pages (from-to)299-328
Number of pages30
JournalJournal of Algebra
Volume503
DOIs
Publication statusPublished - 2018

Keywords

  • algebra
  • graded rings
  • injective modules (algebra)
  • rings (algebra)

Fingerprint

Dive into the research topics of 'Leavitt path algebras : graded direct-finiteness and graded Σ-injective simple modules'. Together they form a unique fingerprint.

Cite this