Abstract
Let be a commutative ring with unity, and a set of nonunit elements is denoted by . The coannihilator graph of , denoted by , is an undirected graph with vertex set (set of all nonzero nonunit elements of ), and is an edge of or , where denotes the principal ideal generated by . In this study, we first classify finite ring , for which is isomorphic to some well-known graph. Then, we characterized the finite ring , for which is toroidal or projective.
1. Introduction
Throughout this study, all rings are commutative ring with unity which is not a field. For , annihilator of , which is denoted by , is defined as . The set of zero-divisor of , the set of nonunit elements of , and the set of unit elements of is denoted by , , and , respectively. For a finite ring , . For , we denote . For any undefined notation or terminology in ring theory, please refer to [1].
Recall that two adjacent vertices in a graph are denoted by . A graph is said to be complete if for each . Also, if , then is denoted by . A graph is said to be bipartite if , such that for (or ), . Also, if , for each and , then is said to be complete bipartite graph. If and , then is denoted by . A graph , which contains a unique cycle, is called a unicycle graph. A graph , which contains no cycle, is called acyclic. A connected acyclic graph is said to be a tree. A graph is called a split graph if , such that , where is an independent set and is a clique. For more details on graph theory, see [2, 3].
The concept of zero-divisor graph was first introduced by Beck [4], where he was mainly interested in the coloring of a commutative ring. In [5], Anderson and Livingston modified the definition of zero-divisor graph and denoted it by , whose vertex set is , and is an edge of ; for more details, see [6–11]. After that, Badawi [12] introduced the annihilator graph of a commutative ring , denoted by , whose vertex set is , and is an edge of . Recently, Afkhami and Khashyarmanesh [13] introduced and studied the cozero-divisor graph of a commutative ring , denoted by with vertex set , and is edge of and . For more details about cozero-divisor graph, see [14–16].
In this study, we study about the coannihilator graph of a commutative ring , which was introduced by Afkhami et al. [17]. It is an undirected graph denoted by with vertex set , and is an edge of or . First, we classify all the finite rings , for which is isomorphic to some well-known graphs. Then, we characterized all the finite rings , for which is toroidal or a projective graph.
We recall some results from [17], which are useful for the proof of the subsequent sections.
Lemma 1 (see [17], Lemma 2.1). Let be a commutative ring. Then, the following hold:(1)For some distinct , is not an edge of (2)If is an edge of , for some distinct , then is an edge of .(3)If is an edge of , for some distinct , then is an edge of .(4)If is an edge of , for some distinct , then is an edge of .
Theorem 1 (see [17], Theorem 5). Let be a commutative ring. Then, is connected and .
Theorem 2 (see [17], Theorem 6). Let be a local commutative ring. Then, is complete.
2. Basic Properties of
In this section, we classify all the finite rings , whose coannihilator graph is isomorphic to some well-known graphs such as complete graph, unicycle graph, tree, or split graph.
Theorem 3. Let be a finite commutative ring. Then, is a complete graph, is local, or for .
Proof. Assume that is complete. Since is finite, , where is local ring for each . If , we are done. Suppose . Suppose that is not a field for some . Consider and , where . Since , in , which contradicts our assumption. Hence, is a field for each .
Suppose . Consider and , where . Since , in , again contradicts our assumption. Hence, for each , which implies that for each .
Converse is clear.
Corollary 1. Let be a finite local commutative ring. Then, the following hold:(1), (2), (3), , , , , , (4), (5), (6), , , , , , , , , , , , , , , , , , ,
Theorem 4. Let be a finite commutative ring. Then, is unicycle, is one of the following rings:
Proof. Assume that is unicycle. Since is finite, , where is the local for each . Suppose . Consider , , , and . Since and , and are two different cycles in , which contradict our assumption. Hence, .
Suppose and is not a field with maximal ideal . Then, there is , such that . Consider , , , and . Since , , , and , and are two distinct cycles in , which contradict our assumption. Hence, and both are fields, which show that . Since is a unicycle, and . Hence, .
Suppose . Then, is a local ring. Thus, is complete by Theorem 2. Since is a unicycle, . Hence, , , , , , , or .
The converse is clear.
Theorem 5. Let be a finite commutative ring. Then, is a tree, is one of the following rings:where is the finite field.
Proof. Assume that is a tree. Since is finite, , where is the local ring for each . Suppose . Consider , , and . Since , is a cycle in , which contradicts our assumption. Hence, .
Suppose and is not a field with maximal ideal . Then, there is , such that . Consider , , and . Since and , is a cycle in , which contradicts our assumption. Hence, and both are fields, which show that . Since is a tree, or . Hence, , where is the finite field.
Suppose . Then, is a local ring. Thus, is complete by Theorem 2. Since is a tree, . This implies that , , , or .
The converse is clear.
Theorem 6 (see [2]). Let be a connected graph. Then, is a split graph, contains no induced subgraph isomorphic to , , and .
Theorem 7. Let be a finite commutative ring with . Then, is split graph, is one of the following rings:where is the finite field.
Proof. Assume that is a split graph. Since is finite, , where is the local ring for each . Suppose . Consider , , , and . Since and , is in , which contradicts our assumption by Theorem 6. Hence, .
Suppose and is not a field with maximal ideal . Then, there is , such that . Consider , , , and , where . Since , , and , is in , which contradicts our assumption by Theorem 6. Hence, and both are fields, which show that . Since is a split graph, or . Hence, , where is the finite field.
Suppose . Then, is a local ring. Thus, is complete by Theorem 2. Since is split, or 3. Hence, , , , , , , or .
The converse is clear.
3. Genus of
If a graph can be embedded on a sphere with handles without crossing itself, then the minimality of such positive integer is called the of the graph , and it is denoted by . Also, if , then the graph is called toroidal.
In this section, we classify all the finite commutative rings for which the coannihilator graph is toroidal.
In order to classify the rings with genus one coannihilator graph, we need the following results, which deal with genus properties of graphs.
Lemma 2. (see [3])(1)Let . Then, In particular, if .(2)Let . Then,In particular, if . Also, , if .
Theorem 8. Let be a finite local commutative ring. Then, is one of the following rings:
Proof. Since is local ring, then by Theorem 2, is a complete graph. Hence, . This completes the proof.
Theorem 9. Let be a finite nonlocal reduced commutative ring. Then, is one of the following rings:
Proof. Since , , , , and , by Lemma 2 and Figure 1.
Now, conversely assume that . Since is a finite nonlocal reduced ring, , where is a field for each and . Suppose . Consider , , , , , , , and . Since , , , , , , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 2, which contradicts our assumption. Hence, .
Suppose and . Consider , , , , , , , , and , where . Since , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 2, which contradicts our assumption. Hence, , for each , which shows that .
Suppose . Then, is a complete bipartite graph of order . Since , by Lemma 2, , , , or .
Theorem 10. Let be a finite nonlocal nonreduced commutative ring. Then, is one of the following rings:
Proof. Suppose or , then the toroidal embedding of is shown in Figure 2.
Conversely, assume that . Since is nonreduced nonlocal finite ring, then the following two cases occur: Case (i): , where is the local ring with for each and . Then, there is , such that for each . Consider , , , , , , , , and , where and . Since , , , , , , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 2, which contradicts our assumption. Case (ii): , where is the local ring with for each , is a field for each , and . If , then we get a contradiction as above. Hence, .Suppose and , such that . Consider , , , , , , , , and . Since , , , , , , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 2, which contradicts our assumption. Hence, , which shows that .
Suppose , then and . Let , such that and . Consider , , , , , , , , and , where . Since , , , , , , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 2, which contradicts our assumption. Hence, .
Suppose , then . Let , such that and . Consider , , , , , , , , and , where . Since and , for and for each , the set induces as a subgraph of . Thus, by Lemma 2, which contradicts our assumption. Hence, , which shows that or .
Suppose and , such that . Consider , , , , , , , , and , where and . Since , , , , , , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 2, which is a contradiction to our assumption. Hence. . From ([17], Theorem 3.1), it is clear that . Hence, .
4. Crosscap of
The crosscap of a graph , denoted by , is the minimum integer , such that can be embedded in , where denotes a sphere with crosscap attached to it. It is clear that , where is the reduction of , and for any subgraph of . A graph is said to be projective if .
In this section, we classify all the finite commutative rings , for which the coannihilator graph is projective. The following results are useful for the proof of subsequent results of this section.
Lemma 3. (see [3])(1)Let . Then,(2)Let . Then,
Lemma 4 (see [3]). Let be a connected graph with vertices and edges. Then,
Theorem 11. Let be a finite commutative ring. Then, is one of the following rings: , , , , or .
Proof. Since , , and , by Lemma 3 and Figure 3, .
Conversely, assume that . Since is finite, , where is the local ring for each . Suppose . Consider , , , , , , , and . Since , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 3, which is a contradiction to our assumption. Hence, . The following three cases occur: Case (i): . Suppose that is not a field with maximal ideal , then there is , such that . Consider , , , , , , , and . Since , , , , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 3, which is a contradiction to our assumption. Hence, is a field for each . Suppose . Consider , , , , , , , and , where . Since , , , , , , , , , , , and , the set induces as a subgraph of . Thus, by Lemma 3, which contradicts our assumption. Hence, for each , which shows that . Case (ii): . If and both are not fields, then by Theorem 10, , a contradiction. Hence, at least one of or must be a field. Suppose is not a field with maximal ideal . Since , by Theorem 10, , , , or . If or , then is planar by ([17], Theorem 8), a contradiction to our assumption. Also, if or , then by Lemma 4 and Figure 2, again a contradiction. Hence, and both are fields, which show that is a complete bipartite graph of order . Since , by Lemma 3, or . Case (iii): . Then, is a local ring. Thus, is complete by Theorem 2. Since , by Lemma 3, . Hence, or .
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.