AI 中文总结
本文研究半群与环的零因子有向图、扩展零因子有向图并引入零化子有向图，建立三类有向图相等的条件，刻画全矩阵环及阿廷非交换环的相关性质。
AI 中文摘要
设S是含零元的半群，本文研究零因子有向图$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}(S)$和扩展零因子有向图$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}_{\text{E}}$，并通过左、右零化子引入零化子有向图$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{AG}}}(S)$。证明了当每个零因子都是幂零元时的直径界，以及$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}_{\text{E}}(S)$的汇点与源点和$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}(S)$的汇点与源点相同。建立了$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}(S)=\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}_{\text{E}}(S)=\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{AG}}}(S)$成立的条件，且当每个零因子都是幂零元或双侧零因子时，对$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{AG}}}(S)$的连通性、直径、围长和顶点度数进行了界定。扩展零因子有向图连通当且仅当零因子有向图连通，且它含有向环当且仅当零因子有向图含该有向环。计算了$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}(S)$、$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}_{\text{E}}(S)$和$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{AG}}}(S)$的 knit 度数。对于含单位元的环R，通过幂零指数和单侧零化子条件刻画了等式$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}_{\text{E}}(R)=\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}(R)$成立的情况；其在域F上的全矩阵环$M_n(F)$中成立当且仅当n=2，且$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{AG}}}(M_n(F))$连通且含一个有向环。此外，对于阿廷非交换环R，证明了$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}(R)$连通当且仅当$\boldsymbol{\rightarrow}\boldsymbol{\boldsymbol{\text{Γ}}}_{\text{E}}(R)$连通当且仅当R的每个单侧单位元都是R的双侧单位元。
英文摘要
Let $S$ be a semigroup with zero. This paper studies the zero-divisor digraph $\overrightarrowΓ(S)$ and the extended zero-divisor digraph $\overrightarrowΓ_{\!E}$, and introduces the annihilator digraph $\overrightarrow{\mathrm{AG}}(S)$ via left and right annihilators. The diameter bound when every zero-divisor is nilpotent, and the sinks and the sources of $\overrightarrowΓ_{\!E}(S)$ being identical to those of $\overrightarrowΓ(S)$ is demonstrated. The conditions in which $\overrightarrowΓ(S)=\overrightarrowΓ_{\!E}(S)=\overrightarrow{\mathrm{AG}}(S)$ holds are established, and the connectedness, diameter, girth, and vertex degrees of $\overrightarrow{\mathrm{AG}}(S)$ are bounded when every zero-divisor is nilpotent or two-sided. The extended zero-divisor digraph is connected if and only if the zero-divisor digraph is connected, and it contains a directed cycle if and only if the zero-divisor digraph does. The knit degrees of $\overrightarrowΓ(S)$, $\overrightarrowΓ_{\!E}(S)$, and $\overrightarrow{\mathrm{AG}}(S)$ are computed. For a unital ring $R$, the equality $\overrightarrowΓ_{\!E}(R)=\overrightarrowΓ(R)$ is characterized by nilpotency indices and one-sided annihilator conditions; it holds for the full matrix ring $M_{n}(F)$ over a field $F$ if and only if $n=2$, and $\overrightarrow{\mathrm{AG}}(M_{n}(F))$ is connected and contains a directed cycle. Moreover, for an artinian noncommutative ring $R$, it was proved that $\overrightarrowΓ(R)$ is connected if and only if $\overrightarrowΓ_{\!E}(R)$ is connected if and only if every one-sided identity element of $R$ is a two-sided identity of $R$.
Comments19 pages, 2 figures
