{
  "abstract": "In the path algebra $KE$, where $K$ is a field and $E$ is a directed graph (or quiver), every non-loop edge can be expressed in the form $arb$, where $a,b,r\\in KE$ and $ab=0$. This fact leads us to the characterization that $KE$ is semicommutative if and only if $E$ contains no non-loop edges. For a ring $R$, let $Z_i(R)=\\{x\\in R: x=arb, a,b,r\\in R, ab=0 \\}$ and call elements of $Z_i(R)$ zero-insertive. It follows that $R$ is semicommutative if and only if $Z_i(R)\\subseteq E(R)$ and weakly semicommutative if and only if $Z_i(R)\\subseteq N(R)$. We establish that every non-unit element of the Leavitt path algebra $L_K(A_2)$ is zero-insertive. If $K$ is a field and $n\\geq 2$ is an integer, then each zero-insertive element of $L_K(A_n)$ is nil-clean if and only if $K\\cong \\mathbb{F}_2$. We call a ring $R$ zero-insertive nil clean (ZINC) if every zero-insertive element is nil clean. We show that the Leavitt path algebra $L_{\\mathbb{F}_2}(A_n)$ is a ZINC ring. Additionally, we investigate the behavior of zero-insertive elements and ZINC rings under various ring extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24439",
  "author_total": 2,
  "authors": [
    {
      "name": "Sanjiv Subba"
    },
    {
      "name": "Tikaram Subedi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2y60gkqmvhssn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24439",
  "fulltext_url": null,
  "id": "2y60gkqmvhssn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24439",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214055601",
  "pdf_url": "https://arxiv.org/pdf/2609.24439",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2y60gkqmvhssn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24439",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24439"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Semicommutativity via Zero-Insertive Elements: Insights from Path and Leavitt Path Algebras",
  "title_text": "Semicommutativity via Zero-Insertive Elements: Insights from Path and Leavitt Path Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/2y60gkqmvhssn",
  "venue": "arXiv",
  "versions": []
}