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Graph characterization of the annihilator ideals of Leavitt path algebras

Lia Vas

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 can 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 characterization 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.

Graph characterization of the annihilator ideals of Leavitt path algebras

Abstract

If is a graph and is a field, we consider an ideal of the Leavitt path algebra of over . We describe the admissible pair corresponding to the smallest graded ideal which contains where the grading in question is the natural grading of by . Using this description, we show that the right and the left annihilators of are equal (which can be somewhat surprising given that may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterization 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 which is equivalent to unital having this property.
Paper Structure (13 sections, 7 theorems, 14 equations)

This paper contains 13 sections, 7 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Prerequisites and preliminaries
  3. Graded rings
  4. Graphs and Leavitt path algebras
  5. Ideals and graded ideals of a Leavitt path algebra
  6. Annihilator ideals
  7. TEXT-saturation of a hereditary set
  8. Graded envelope and annihilator ideals
  9. Quasi-Baer Leavitt path algebras
  10. Graded quasi-Baer property
  11. Quasi-Baer Leavitt path algebras
  12. Relation to Baer and Rickart Leavitt path algebras
  13. Quasi-Baer *-rings

Key Result

Theorem 3.1

Let $I$ be an ideal of $L_K(E)$ and let $(H,S),$$C,$ and $P$ be such that $I=I((H,S), C, P).$ Let $C^0$ be the set of vertices on cycles which are in $C$ and let Then $T\subseteq B_G,$$(H,S)\leq (G,T),$ and

Theorems & Definitions (16)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • ...and 6 more