Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-09T00:33:43.889Z Has data issue: false hasContentIssue false
  • English
  • Français

GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  15 February 2024

LIA VAŠ Open the ORCID record for LIA VAŠ [Opens in a new window]
LIA VAŠ*
Affiliation:
Department of Mathematics, Saint Joseph’s University, Philadelphia, PA 19131, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$. Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property.

Keywords

annihilatorgradedidealLeavitt path algebragraded envelopequasi-Baer

MSC classification

Secondary: 16D25: Ideals 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 10
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abrams, G., Ara, P. and Siles Molina, M., Leavitt Path Algebras, Lecture Notes in Mathematics, 2191 (Springer, London, 2017).CrossRefGoogle Scholar
Birkenmeier, G. F., Kim, J. Y. and Park, J. K., ‘Self-adjoint ideals of Baer $\ast$ -rings’, Comm. Algebra 28(9) (2000), 4259–4268.CrossRefGoogle Scholar
Clark, W. E., ‘Twisted matrix units semigroup algebras’, Duke Math. J. 34 (1997), 417–424.Google Scholar
Gil Canto, C., Martín Barquero, D. and Martín Gonzáles, C., ‘Invariants ideals in Leavitt path algebras’, Publ. Mat. 66 (2022), 541–569.CrossRefGoogle Scholar
Gonçalves, D. and Royer, D., ‘A note on the regular ideals of Leavitt path algebras’, J. Algebra Appl. 21(11) (2022), Article no. 2250225.CrossRefGoogle Scholar
Hazrat, R., ‘Leavitt path algebras are graded von Neumann regular rings’, J. Algebra 401 (2014), 220–233.CrossRefGoogle Scholar
Hazrat, R. and Vaš, L., ‘Baer and Baer $\ast$ -ring characterizations of Leavitt path algebras’, J. Pure Appl. Algebra 222(1) (2018), 39–60.CrossRefGoogle Scholar
Vaš, L., ‘Annihilator ideals of graph algebras’, J. Algebraic Combin. 58(2) (2023), 331–353.CrossRefGoogle Scholar