论文标题
图形代数的歼灭者理想
Annihilator ideals of graph algebras
论文作者
论文摘要
如果$ i $是环$ r $的（双面）理想，我们让$ \ operatotorname {ann} _l（i）= \ {r \ in r \ mid ri = 0 \}，$ $ \ operatorname {ann} $ \ operatoTorname {ann}（i）= \ operatatorName {ann} _l（i）\ cap \ operatatorName {ann} _r（i）$ be左，右，右和双nihihilators。如果$ i = \ operatorName {ann}（j）$对于某些理想的$ j $（等效地，$ \ propatatorName {ann}（\ permatorAtorName {ann}（ann}（i）），则理想的$ i $是nihihilator的理想。我们研究Leavitt Path代数的歼灭者理想和图形$ C^*$ - 代数。 让$ l_k（e）$是field $ k上的图$ e $的leavitt路径代数。我们注意到，$ \ operatorAtorname {ann} _l（i）$和$ \ operatatorName {ann} _r（i）$也被分级。 For a graded ideal $I,$ we describe $\operatorname{ann}(I)$ in terms of the properties of a pair of sets of vertices of $E,$ known as an admissible pair, which naturally corresponds to $I.$ Using such a description, we present properties of $E$ which are equivalent with the requirement that each graded ideal of $L_K(E)$ is an annihilator ideal.我们表明，$ e $的相同属性也与以下每个条件相同：（1）$ l_k（e）$的分级理想的晶格是布尔式代数； （2）$ c^*（e）$的每个封闭规数的理想是an灭者的理想； （3）$ c^*（e）$的封闭规数理想的晶格是布尔代数。此外，我们介绍了$ e $的属性，这些属性与以下每个条件相同：（1）$ l_k（e）$的每个理想都是an灭者的理想； （2）$ l_k（e）$的理想晶格是布尔代数； （3）$ c^*（e）$的每个封闭理想都是歼灭者的理想； （4）$ c^*（e）$的封闭理想的晶格是布尔代数。
If $I$ is a (two-sided) ideal of a ring $R$, we let $\operatorname{ann}_l(I)=\{r\in R\mid rI=0\},$ $\operatorname{ann}_r(I)=\{r\in R\mid Ir=0\},$ and $\operatorname{ann}(I)=\operatorname{ann}_l(I)\cap \operatorname{ann}_r(I)$ be the left, the right and the double annihilators. An ideal $I$ is said to be an annihilator ideal if $I=\operatorname{ann}(J)$ for some ideal $J$ (equivalently, $\operatorname{ann}(\operatorname{ann}(I))=I$). We study annihilator ideals of Leavitt path algebras and graph $C^*$-algebras. Let $L_K(E)$ be the Leavitt path algebra of a graph $E$ over a field $K.$ If $I$ is an ideal of $L_K(E),$ it has recently been shown that $\operatorname{ann}(I)$ is a graded ideal (with respect to the natural grading of $L_K(E)$ by $\mathbb Z$). We note that $\operatorname{ann}_l(I)$ and $\operatorname{ann}_r(I)$ are also graded. For a graded ideal $I,$ we describe $\operatorname{ann}(I)$ in terms of the properties of a pair of sets of vertices of $E,$ known as an admissible pair, which naturally corresponds to $I.$ Using such a description, we present properties of $E$ which are equivalent with the requirement that each graded ideal of $L_K(E)$ is an annihilator ideal. We show that the same properties of $E$ are also equivalent with each of the following conditions: (1) The lattice of graded ideals of $L_K(E)$ is a Boolean algebra; (2) Each closed gauge-invariant ideal of $C^*(E)$ is an annihilator ideal; (3) The lattice of closed gauge-invariant ideals of $C^*(E)$ is a Boolean algebra. In addition, we present properties of $E$ which are equivalent with each of the following conditions: (1) Each ideal of $L_K(E)$ is an annihilator ideal; (2) The lattice of ideals of $L_K(E)$ is a Boolean algebra; (3) Each closed ideal of $C^*(E)$ is an annihilator ideal; (4) The lattice of closed ideals of $C^*(E)$ is a Boolean algebra.
