论文标题
零划分图和连续函数环的comaximal图，可计数范围
Zero-divisor graph and comaximal graph of rings of continuous functions with countable range
论文作者
论文摘要
在本文中，两个外部不同的图，即，零除数图$γ（c_c（x））$和comaximal Graph $γ_2^{'}（c_c（x））$的ring $ c_c（x）$ c_c（x）$的所有真实连续函数的可计数范围内定义了任何可计数的$ hausdorfff zere $ dimensdorfff zere $ dimensdorfff zere $ dimensdorfff。据观察，这两个图表表现出相似之处，就直径，腰围，连通性，三角测量或过度对三角测量而言。很关心。但是，该研究表明，$ c_c（x）$的中间环$ a_c（x）$的零除数图$γ（a_c（x））$是互补的。此外，$γ（c_c（x））$仅在补充其子图$γ（a_c（x））$时才得到补充。另一方面，仅当且仅当其过度的$ c（x）$的comaximal图被互补并且已知后一个图表时，只有$ x $是$ p $ -p $ - 空格时，就知道后一个图是互补的。的确，对于大量的空间（即，对于完全正常的，强的尺寸空间（不是P空间）），$γ（C_C（X））$和$γ_2^{'}（c_c（x））$被认为是非iSomorphic的。适当地定义图的商，它可以用来确定对于离散的空间$ x $，$γ（c_c（x））$（= $γ（c（x））$）和$γ_2^{'}（c_c（x）$（= $γ_2^{'}'}（c_c（x））$ countic，If Countic，If Countic，If Countic，在连续假设的假设下，该结果的相反也被证明是正确的。
In this paper, two outwardly different graphs, namely, the zero divisor graph $Γ(C_c(X))$ and the comaximal graph $Γ_2^{'}(C_c(X))$ of the ring $C_c(X)$ of all real-valued continuous functions having countable range, defined on any Hausdorff zero dimensional space $X$, are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness. are concerned. However, the study reveals that the zero divisor graph $Γ(A_c(X))$ of an intermediate ring $A_c(X)$ of $C_c(X)$ is complemented if and only if the space of all minimal prime ideals of $A_c(X)$ is compact. Moreover, $Γ(C_c(X))$ is complemented when and only when its subgraph $Γ(A_c(X))$ is complemented. On the other hand, the comaximal graph of $C_c(X)$ is complemented if and only if the comaximal graph of its over-ring $C(X)$ is complemented and the latter graph is known to be complemented if and only if $X$ is a $P$-space. Indeed, for a large class of spaces (i.e., for perfectly normal, strongly zero dimensional spaces which are not P-spaces), $Γ(C_c(X))$ and $Γ_2^{'}(C_c(X))$ are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilised to establish that for a discrete space $X$, $Γ(C_c(X))$ (= $Γ(C(X))$) and $Γ_2^{'}(C_c(X))$ (= $Γ_2^{'}(C(X))$) are isomorphic, if $X$ is atmost countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.
