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Directed Graphs of Commutative Rings Rose-Hulman Undergraduate Mathematics Journal Volume 14 Issue 2 Article 11 Directed Graphs of Commutative Rings Seth Hausken University of St. Thomas, [email protected] Jared Skinner University of St. Thomas, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Hausken, Seth and Skinner, Jared (2013) "Directed Graphs of Commutative Rings," Rose-Hulman Undergraduate Mathematics Journal: Vol. 14 : Iss. 2 , Article 11. Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss2/11 Rose- Hulman Undergraduate Mathematics Journal Directed Graphs of Commutative Rings Seth Hausken a Jared Skinnerb Volume 14, no. 2, Fall 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a Email: [email protected] University of St. Thomas b http://www.rose-hulman.edu/mathjournal University of St. Thomas Rose-Hulman Undergraduate Mathematics Journal Volume 14, no. 2, Fall 2013 Directed Graphs of Commutative Rings Seth Hausken Jared Skinner Abstract. The directed graph of a commutative ring is a graph representation of its additive and multiplicative structure. Using the mapping (a; b) ! (a + b; a · b) one can create a directed graph for every finite, commutative ring. We examine the properties of directed graphs of commutative rings, with emphasis on the informa- tion the graph gives about the ring. Acknowledgements: We would like to acknowledge the University of St. Thomas for making this research possible and Dr. Michael Axtell for his guidance throughout. Page 168 RHIT Undergrad. Math. J., Vol. 14, no. 2 1 Introduction When studying group theory, especially early on in such an exploration, Cayley tables are used to give a simple visual representation to the structure of the group. Cayley tables can also be used when studying rings, but since rings have both an additive and multiplicative structure, two Cayley tables are necessary to fully represent a ring. It is desirable to create a visual representation of a ring that maintains the structure while being a single representation instead of two. We turn to graph theory to create a directed graph representation of a ring, as first proposed by Lipkovski in [8]. Definition By the directed graph, or digraph for brevity, of R, denoted Ψ(R), we mean the graph with V (Ψ(R)) = R × R, and for distinct (a; b); (c; d) 2 R × R, there is a directed edge, denoted (a; b) ! (c; d), connecting (a; b) to (c; d) if and only if a + b = c and a · b = d. Example 1.1 The digraph of Z4 consists of vertices with entries from Z4 with addition and multiplication modulo 4 [See Figure 1]. 1, 1 2, 3 1, 2 3, 0 3, 1 3, 2 0, 3 2, 1 1, 3 3, 3 1, 0 0, 1 2, 0 0, 2 2, 2 0, 0 Figure 1: Ψ(Z4) The digraph of a ring holds the additive and multiplicative structure of the ring in a simpler, more compressed fashion. Unlike other common graph representations of rings, such as the zero-divisor graph [1], the digraph attempts to retain all the elements and structure of the ring while still being fairly straight forward. Since the digraph does retain the entire RHIT Undergrad. Math. J., Vol. 14, no. 2 Page 169 structure of the ring, the digraph should contain information about algebraic structures, such as ideals and zero-divisors. 2 Background In this paper, we use concepts from both ring theory and graph theory. Both these topics are far too dense to delve into with great detail here. Instead, we define some of the basic concepts needed for this paper, then further define concepts as they become relevant. For further explanation of ring theory, see [2], [5], [6], [4]. For further explanation of graph theory, see [3]. Definition [4] A ring is a set with two binary operations, addition and multiplication, such that for all a, b, and c 2 R: 1. a + b = b + a 2. (a + b) + c = a + (b + c) 3. There is an additive identity 0 in R. That is, there is an element 0 in R such that a + 0 = a for all a 2 R. 4. There is an element −a in R such that a + (−a) = 0. 5. a(bc) = (ab)c 6. a(b + c) = ab + ac and (b + c)a = ba + ca A given ring R is called commutative if multiplication in the ring is commutative. That is, for all a; b 2 R, ab = ba. A given ring R is said to have identity if there exists 1 2 R such that a · 1 = a for all a 2 R. By a ring, we generally mean a commutative ring with identity, denoted R, unless otherwise specified. While it is only feasible to graph finite rings, results are given for all rings, unless otherwise specified. The notion of a subring is exactly as one would expect, a ring that is contained within another ring and has the same structure as the containing ring. Next, we define a special type of subring called an ideal. Definition [4] A subring of a ring R is called an ideal of R is for every r 2 R and every a 2 A, ra and ar are in A. By an ideal, we mean a proper ideal of a ring, that is an ideal that is neither the trivial ring nor the entire ring R. Next, we define a zero-divisor, which is a particular type of element of a ring with peculiar and interesting properties. Page 170 RHIT Undergrad. Math. J., Vol. 14, no. 2 Definition A zero-divisor is an element a of a commutative ring R such that there is a nonzero element b in R such that ab = 0. Note that this implies that 0 is a zero-divisor in every non-trivial ring. Some texts define zero-divisors as strictly nonzero elements, but we adopt the above definition from [2] and [6]. We adopt the notation Z(R) for the set of all zero-divisors of a commutative ring from [6]. The set of units of R is denoted U(R). The remaining definitions are from graph theory. Definition [3] A graph G is an ordered pair of disjoint sets (V; E) such that E is a subset of the set V 2 of unordered pairs of V . The set V is the set of vertices of G and the set E is the set of edges of G. An edge (x; y) joins vertices x and y. For a graph G, the set of vertices is denoted V (G) and the set of edges is denoted E(G). We say G0 is a subgraph of G if V (G0) ⊆ V (G) and E(G0) ⊆ E(G). If G0 contains all edges of G that join two vertices in V (G0), then G0 is an induced subgraph. For our purposes, all subgraphs will be induced, so we will use the term subgraph to mean induced subgraph. The digraph of an ideal I of a ring R, denoted Ψ(I), is a subgraph of Ψ(R) where V (Ψ(I)) = I×I. Definition [3] If the edges of a graph G are ordered pairs of vertices, then G is called a directed graph. An ordered pair (a; b) is directed from a to b. Definition [3] If (a1; b1); (a2; b2);:::; (an; bn) are vertices in Ψ(R), then (a1; b1) (a2; b2) ··· (an; bn) denotes a walk in Ψ(R) from vertex (a1; b1) to vertex (an; bn), where (ai; bi) is adjacent to (ai+1; bi+1) and (ai−1; bi−1) for 1 ≤ i ≤ n − 1 and denotes a connection between two vertices without regard to direction. If (a1; b1); (a2; b2);:::; (an; bn) are distinct, then the walk is a path. For directed graphs, it becomes necessary to make a distinction between a walk (path) and a directed walk (directed path). Definition If (a1; b1); (a2; b2);:::; (an; bn) are vertices in Ψ(R), then (a1; b1) ! (a2; b2) ! ···! (an; bn) denotes a directed walk in Ψ(R) from vertex (a1; b1) to vertex (an; bn), where (ai; bi) is directionally adjacent to (ai+1; bi+1) for 1 ≤ i ≤ n−1. If (a1; b1); (a2; b2);:::; (an; bn) are distinct, then the directed walk is a directed path. Definition A connected graph is one in which there is a path between two distinct vertices. A connected component of a graph G, denoted C, is a maximal connected subgraph of G. RHIT Undergrad. Math. J., Vol. 14, no. 2 Page 171 3 Paths We begin by determining the behavior of paths and walks in the digraph of a ring. It is necessary to begin with such results as they are used later when studying the algebraic structures represented in digraphs. Definition Let (a; b); (x; y) 2 Ψ(R). Suppose (a; b) !···! (x; y), then (x; y) is said to be downstream from (a; b), and (a; b) is said to be upstream from (x; y). Definition If (x; y) ! (z; w), we say that (x; y) points to (z; w). Definition Given a vertex v, the number of vertices that point at v is the incoming degree of v. The number of vertices that v points to is the outgoing degree of v. It should be noted that because (x; y) ! (x + y; xy), the outgoing degree of (x; y) will always be one. Definition A cycle of length n is a directed path such that (x1; y1) ! (x2; y2) !···! (xn; yn) ! (x1; y1).
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