通过零插入元素的半交换性：来自路径代数和Leavitt路径代数的洞见

中文摘要 AI 辅助

在路径代数$KE$中，其中$K$是一个域，$E$是一个有向图（或箭图），每条非环边都可以表示为$arb$的形式，其中$a,b,r\in KE$且$ab=0$。这一事实引导我们得到刻画：$KE$是半交换的当且仅当$E$不包含非环边。对于环$R$，令$Z_i(R)=\{x\in R: x=arb, a,b,r\in R, ab=0 \}$，并将$Z_i(R)$中的元素称为零插入元素。由此可知，$R$是半交换的当且仅当$Z_i(R)\subseteq E(R)$，且$R$是弱半交换的当且仅当$Z_i(R)\subseteq N(R)$。我们证明了Leavitt路径代数$L_K(A_2)$的每个非单位元素都是零插入的。若$K$是一个域且$n\geq 2$为整数，则$L_K(A_n)$的每个零插入元素都是幂清洁的当且仅当$K\cong \mathbb{F}_2$。我们将环$R$称为零插入幂清洁（ZINC）环，如果每个零插入元素都是幂清洁的。我们证明了Leavitt路径代数$L_{\mathbb{F}_2}(A_n)$是ZINC环。此外，我们研究了在各种环扩张下零插入元素和ZINC环的行为。

英文摘要

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.

发表机构

• School of Applied Sciences UPES（UPES应用科学学院）
• Department of Mathematics National Institute Of Technology Meghalaya（梅加拉亚邦国立技术学院数学系）

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