Semicommutativity via Zero-Insertive Elements: Insights from Path and Leavitt Path Algebras
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∈ 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 Zi(R)={x∈ R: x=arb, a,b,r∈ R, ab=0 } and call elements of Zi(R) zero-insertive. It follows that R is semicommutative if and only if Zi(R)⊆ E(R) and weakly semicommutative if and only if Zi(R)⊆ N(R). We establish that every non-unit element of the Leavitt path algebra LK(A2) is zero-insertive. If K is a field and n≥ 2 is an integer, then each zero-insertive element of LK(An) is nil-clean if and only if K≅ 𝔽2. 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 L𝔽2(An) is a ZINC ring. Additionally, we investigate the behavior of zero-insertive elements and ZINC rings under various ring extensions.
Related
- On the existence of divisible non-injective modules over Leavitt path algebras
- Growth functions of algebras and an application to Leavitt path algebras
- On some relationships between Leavitt path algebras and Brauer configuration algebras
- Graded classification of Leavitt path algebras in terms of strong shift equivalence
- Baer-Type Ring Characterizations of Steinberg Algebras, with Applications to Leavitt Path Algebras
BibTeX
@misc{subba2026semicommutativity,
title = {{Semicommutativity via Zero-Insertive Elements: Insights from Path and Leavitt Path Algebras}},
author = {Sanjiv Subba and Tikaram Subedi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24439},
eprint = {2609.24439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24439},
}
RIS
TY - UNPB
TI - Semicommutativity via Zero-Insertive Elements: Insights from Path and Leavitt Path Algebras
AU - Sanjiv Subba
AU - Tikaram Subedi
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24439
UR - https://arxiv.org/abs/2609.24439
AB - 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∈ 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∈ R: x=arb, a,b,r∈ 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)⊆ E(R) and weakly semicommutative if and only if Z_i(R)⊆ 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≥ 2 is an integer, then each zero-insertive element of L_K(A_n) is nil-clean if and only if K≅ 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_𝔽_2(A_n) is a ZINC ring. Additionally, we investigate the behavior of zero-insertive elements and ZINC rings under various ring extensions.
ER -