The distant graph of the projective line over a finite ring with unity

We discuss the projective line $\mathbb{P}(R)$ over a finite associative ring with unity. $\mathbb{P}(R)$ is naturally endowed with the symmetric and anti-reflexive relation"distant". We study the graph of this relation on $\mathbb{P}(R)$ and classify up to isomorphism all distant graphs $G(R, \Delta)$ for rings $R$ up to order $p^5$, $p$ prime.


Introduction
The aim of this paper is to characterize the distant graph G(R, ∆) of the projective line over any finite ring R. It is an undirected, connected graph with the degree of a vertex equal to |R|. The starting point of our investigation is showing the connection between this graph and the distant graph G(R/J, ∆ J ) of the projective line over the factor ring R/J, where J is the Jacobson radical of R. To this end we use, introduced by Blunck and Havlicek in [6], an equivalence relation, called radical parallelism, on the set of points of the projective line, which determines the interdependence between P(R) and P(R/J). Next we describe the graph G(R/J, ∆ J ). Using structures theorems [2] on finite rings with unity we get that the graph G(R/J, ∆ J ) is isomorphic to the tensor product of the distant graphs arising from projective lines whose underlying rings are full matrix rings over finite fields. The projective line over any full matrix ring M n (q), i.e. the ring of n×n matrices over the finite field F (q) of order q, is in bijective correspondence with the Grassmannian G (n, 2n, q) of n-dimensional subspaces of a 2n-dimensional vector space over F (q). Then we describe G(M 2 (q), ∆) for any prime power q and we give representatives of two classes of partitions of G(M 2 (2), ∆) on a sum of vertex-disjoint maximal cliques. We also make use of these partitions to show a simple construction of projective space of order 2, described by Hirschfeld in [16] in a completely different way. The question still unanswered is, whether a partition of G M n (q), ∆) on a sum of vertex-disjoint maximal cliques exists for any n, q. However, this partition of G(R, ∆) for any finite ring R such that R/J is isomorphic to the direct product of n copies of F (q) is done in the present study, in particular the distant graph of the projective line over any ring of lower triangular matrices over F (q) . Using the classification of finite rings from [13], [11], we find all nonisomorphic distant graphs G(R, ∆) for rings R up to order p 5 , p prime, in the last section. We also describe the graph G(R, ∆) in the case of an arbitrary local ring R.

Preliminaries
Throughout this paper we shall only study finite associative rings with 1 (1 = 0). Consider the free left module 2 R over a ring R.
is a left cyclic submodule of 2 R. If the equation (ra, rb) = (0, 0) implies that The general linear group GL 2 (R) acts in natural way (from the right) on the free left R-module 2 R and this action is transitive.
The projective line over R is the orbit of the free cyclic submodule R(1, 0) under the action of GL 2 (R).
In other words, the points of P(R) are those free cyclic submodules R(a, b) ∈ 2 R which possess a free cyclic complement, i.e. they are generated by admissible pairs (a, b). We recall that a pair (a, b) ∈ 2 R is unimodular, if there exist x, y ∈ R such that ax + by = 1.
It is known that if R is a ring of stable rank 2, then admissibility and unimodularity are equivalent and R is Dedekind-finite [5,Remark 2.4]. Rings that are finite or commutative satisfy this property, so in case of such rings, the projective line can be described by using unimodular or admissible pairs interchangeably.
A wealth of further references is contained in [15], [9].
[6] The point set P(R) is endowed with the symmetric and anti-reflexive relation distant which is defined via the action of GL 2 (R) on the set of pairs of points by It means that Moreover, The next relation on P(R) is connected with the Jacobson radical of R, denoted by J. It is that two-sided ideal which is the intersection of all the maximal right (or left) ideals of R. Namely, in [6] A. Blunck and H.
Havlicek introduced an equivalence relation in the set of pairs of non-distant points called radical parallelism ( ) as follows: where ∆(R(a, b)) is the set of those points of P(R) which are distant to R(a, b) ∈ P(R). In this case we say that a point R(a, b) ∈ P(R) is radically parallel to a point R(c, d) ∈ P(R). The canonical epimorphism R → R/J sends any a ∈ R to a → a + J =: a.
where ∆ J denotes the distant relation on P(R/J).
Therefore, the radical parallelism relation determines the connection between projective lines P(R) and P(R/J).
Since the point set P(R) is endowed with the distant relation, we can consider P(R) as the set of vertices V G(R, ∆) of the distant graph G(R, ∆), i.e. the undirected graph of the relation ∆. Its vertices are joined by an edge if, and only if, they are distant. This graph is connected and its diameter is less or equal 2 [15, 1.4

.2. Proposition].
One of the basic concepts of graph theory is that of a clique. A clique in an undirected graph G is a subset of the vertices such that every two distinct vertices comprise an edge, i.e. the subgraph of G induced by these vertices is complete. A maximum clique of a graph G is a clique, such that there is no clique in G with more vertices. A maximal clique is a clique which is not properly contained in any clique. All maximal cliques in G(R, ∆) has the same number of vertices, denoted by ω G(R, ∆) , and at the same time, they are maximum cliques.
To describe maximal cliques of the distant graph we make use of the following definitions.
Definition 3. An (n−1)-spread in the (2n−1)-dimensional projective space P G(2n − 1, q) over the finite field with q elements F (q) is a set of (n − 1)dimensional subspaces such that each point of P G(2n − 1, q) is contained in exactly one element of this set.
In our considerations we will often use the following fact about the direct product of projective lines.
where ∆ i stands for the distant relation on P(R i ).
In [1] was pointed out that in another way to state this is to say what means that the graph G(R, ∆) is the tensor product of the graphs

Construction of the distant graph on the projective line
In order to describe the distant graph G(R, ∆) of the projective line over a ring R we show the connection between this graph and the distant graph G(R/J, ∆ J ) of the projective line over the factor ring R/J. Next we find the graph G(R/J, ∆ J ). The points of P(R/J) are in one-one correspondence with the equivalence classes of the radical parallelism relation on P(R). Each of these comprises |J| elements. See [6] for more details. Write ). Then the graph G(R, ∆) is uniquely determined by the Remark 1. For example, if R = T (2) is the ring of ternions over the field F (2), then the projective line over T (2)/J has a distant graph which is depicted in Figure 1 Proof. This follows from Remark 1 and the the fact that Φ −1 R/J(a, b) containes exactly |J| points for any (R/J(a, b).
Suppose that the above assumptions are satisfied. We have We first give the proof for the case n = 2. Without loss of generality, assume that s 1 s 2 . Let (k l 1 , k l 2 ), l 1 = 1, . . . , s 1 , l 2 = 1, . . . , s 2 , be vertices of By the definition of the tensor product of graphs G 1 , G 2 we get that vertices (k l 1 , k l 2 ), (k l 1 , k l 2 ) in G 1 × G 2 are joined by an edge if, and only if, k l 1 and k l 1 , k l 2 and k l 2 comprise edges in G 1 , G 2 respectively. All vertices k l i , k l i are elements of the clique K t i , and so they are joined by an edge if, and only if, l i = l i . Therefore vertices writed down in rows of the above sets K t 1 × K t 2 are maximal cliques in G 1 × G 2 . If K t 1 , K t 1 ∈ Kt 1 and t 1 = t 1 , then there is no any vertex in K t 1 × K t 2 forming edges with all elements of some vertexdisjoint maximal clique of K t 1 × K t 2 . Thus m 1 is a sum of m 1 m 2 s 2 vertex-disjoint maximal cliques with s 1 elements. Applying the induction we get the claim.
Theorem 2. Let R be a finite ring such that R/J is isomorphic to R 1 ×· · ·× R n and V G(R i , ∆ i ) is a sum of m i vertex-disjoint maximal cliques K i with |K i | = s i for all i = 1, . . . , n and let min{s i ; i = 1, . . . , n} = s. There exists a partition of G(R, ∆) on a sum of m 1 ···mns 1 ···sn s |J| vertex-disjoint maximal cliques with ω G(R, ∆) = s.
Proof. In view of Theorem 1, we have G(R/J, ∆ J ) G(R 1 , ∆ 1 )×G(R 2 , ∆ 2 )× · · · × G(R n , ∆ n ). By Proposition 2, V G(R/J, ∆ J ) is a sum of m 1 · · · m n s 1 · · · s n vertexdisjoint maximal cliques with ω G(R, ∆) = s. Proposition 1 now yields to desired claim. Corollary 1. Let R be a ring such that R/J is isomorphic to the direct product of n copies of F (q). There exists a partition of the distant graph G(R, ∆) on a sum of (q + 1) n−1 |J| vertex-disjoint maximal cliques with ω G(R, ∆) = q + 1. The ring of lower triangular n × n matrices over the field F (q) is one example of such rings and |J| = q where R i , R σ(i) are full matrix rings over finite fields, with a permutation σ of {1, 2, . . . , n} such that α i : R i → R σ(i) is an isomorphism or an antiisomorphism.
Any finite ring with identity is semiperfect. By the structure theorem of such rings [2] R/J is artinian semisimple and idempotents lift modulo J. Hence it has a unique decomposition into a direct product of simple rings: According to Theorem 1 we get: Any simple ring R i is isomorphic to a full matrix ring M n i (q i ) over the finite field with q i elements: It follows then that the description of the projective line over any finite ring can be based on the projective line over the full matrix ring P(M n (q)). There is a bijection between P(M n (q)) and the Grassmannian G (n, 2n, q), i.e. the set of all n-dimensional subspaces of V (2n, q) [4,2.4 Theorem.]. Consequently, any point of P(M n (q)) can be expressed by using of a basis of the corresponding n-dimensional subspace of V (2n, q). The point (q n1 q n2 . . . q nn q n1 q n2 . . . q nn ) .
The distant graph of the projective line over the full matrix ring G(M n (q), ∆), is isomorphic to the graph on G (n, 2n, q) whose vertex set is G (n, 2n, q) and whose edges are pairs of complementary subspaces X, Y ∈ G (n, 2n, q): Another graph on G (n, 2n, q) is the well known Grassmann graph, which has the same set of vertices as the distant graph but X, Y ∈ G (n, 2n, q) form an edge, whenever both X and Y have codimension 1 in X + Y , i.e. they are adjacent (in symbols: ∼ ): G(M n (q), ∆) can be described using the notion of the Grassmann graph [7, Theorem 3.2]. These graphs have been thoroughly investigated by different authors (see for example [18]), however, this special case of the Grassmann graph G (n, 2n, q) is not characterized. We can give the number of vertices of G(M n (q), ∆) (cf. [14, p. 920]), i.e. the number of n-dimensional subspaces of V (2n, q): The degree of a vertex v ∈ G(M n (q), ∆) is equal to the number of ndimensional subspaces of V (2n, q) that are disjoint to any n-dimensional subspace: It means that deg(v) = |M n (q)| and generally if v ∈ G(R, ∆) then deg(υ) = |R|, which is also due to the fact that GL 2 (R) acts transitively on P(R). Maximal cliques in G(M n (q), ∆) correspond to n-spreads in the 2n-dimensional vector space over F . It is known that such a n-spread containes q n + 1 ndimensional vector subspaces. Any partition of the distant graph G(M n (q), ∆) on a sum of vertex-disjoint maximal cliques corresponds to an n-parallelism of the vector space V (2n, q). Therefore and on account of [3, Theorem 1.], which has been also proved (independently) by Denniston [12], there exists a partition of the distant graph G(M 2 (q), ∆) on a sum of q 2 + q + 1 vertex-disjoint maximal cliques with ω G(M 2 (q), ∆) = q 2 + 1 for any q.
We pay attention now to the distant graph G(M 2 (2), ∆) which has 35 vertices. We can identify the graph G M n (q), ∆) and the corresponding Grassmannian G (n, 2n, q). Then all automorphisms of the distant graph G(M 2 (2), ∆) are linear or superpositions of linear with the automorphisms defined by duality and annihilator mapping; see [19]. Automorphisms of the first type fix the two conjugacy classes of partitions and these of the second type exchange them.
Below we write down one partition from each conjugacy class. In both tables the seven members of the partition are maximal cliques of size five, which are labelled as I, II, . . . , VII. Thereby each point of the graph G(M 2 (2), ∆) is described in terms of two basis vectors of its corresponding subspace in G (2, 4, 2  We study now cliques formed by vertices of any two maximal cliques of the first partition (Table 1). We see that there exists exactly one maximum clique with four elements for any two different maximal cliques. As an example, we show edges formed by vertices of cliques I and II (Figure 2). Edges comprised by vertices of maximum clique are represented by thicker line. So, for any two of three vertex-disjoint maximal cliques we have one maximum clique and we checked that three such maximum cliques are of two distinct kinds: either any two of them have one common vertex (Figure 3) or they are pairwise disjoint (Figure 4). The same result can be drawn for the second partition. By direct verification we found that vertex-disjoint maximal cliques are points of the projective plane of order 2. As lines of this plane we take triples of vertex-disjoint maximal cliques of the second kind ( Figure 5). There is no proof of the existence of a partition of any graph G(M n (q), ∆). But this problem is well known as an n-parallelism in combinatorial design. Sarmiento in [20] described the partition of the design corresponding to that of G(M 3 (2), ∆).

The classification of distant graphs
We start with a characterization of the distant graph of the projective line over any local ring.
Theorem 5. Let R be a local ring. There exists a partition of the distant graph G(R, ∆) on a sum of |J| vertex-disjoint maximal cliques with ω G(R, ∆) = |R/J| + 1.
Proof. If R is local, then J is the maximal ideal of R, R/J is a field, and so G(R/J, ∆ J ) is a complete graph with |R/J| + 1 vertices. According to the connection between G(R/J, ∆ J ) and G(R, ∆) described in section 3, taking into account Remark 1 we obtain that vertices R(a i , b i ) of G(R, ∆) corresponding to the vertex R/J(a, b) of G(R/J, ∆ J ) are not joined by an edge, while they form an edge with any other vertex of G(R, ∆). This finishes the proof.
Let now v j i , u l k be vertices of G(R, ∆) and let V G(R, ∆) , E G(R, ∆) be the sets of vertices and edges of this graph respectively. We described G(R, ∆) explicitly in case of a local ring R: Any finite commutative ring is the direct product of local rings [17,VI.2]. Thus the distant graph of the projective line over any finite commutative ring is known by the above and Theorem 1. Every finite ring is isomorphic to the direct product of rings of prime power order [17, I.1]. Hence the distant graph of the projective line over a finite ring can be also described as the tensor product of the distant graphs of the projective lines over rings of prime power order. We classify below distant graphs G(R, ∆), where R is an indecomposable ring up to order p 5 , p prime. We use some facts and the notations that were established in [11]. Namely, any finite ring can be represented as R = S ⊕ M , where S = m i=1 R i , R i are primary rings and M is a bimodule over the ring S. M is also an additive subgroup of J, so M ⊂ J and we thus get R/J S/J. In the case of p, p 2 for any p we get complete graphs of order p + 1, p + 2 and the graph G(R, ∆), where R is local and |R| = p 2 , |J| = p. If |R| = p 3 then we have three graphs G(R, ∆) for any p: complete graph of order p 3 + 1, the graph where R is local and |J| = p 2 and the graph of the projective line over the ring of lower triangular 2 × 2-matrices over a field F (p), which is a sum of p 2 + p vertex-disjoint maximal cliques with ω G(R, ∆) = p + 1.
Theorem 6. Let R be an indecomposable ring of order p 4 , p prime. There are exactly five nonisomorphic graphs G(R, ∆) for any p. These are: 1. the complete graph of order p 4 + 1; 2. the graph on the projective line over local ring with |J| = p 2 ; (b) F (p 2 ) × F (p) or F (p) × F (p 2 ). It is necessary to explain how these two ring parts represent two distinct rings. And that is, they have different module parts: F (p 2 ) M F (p) and F (p) M F (p 2 ) respectively. For all these rings R/J = F (p) × F (p 2 ).
The connection between G(R, ∆) and G(R/J, ∆ J ) and Theorems 1 and 5 now completes the proof.
When this paper was finished we became aware of the recent preprint by Silverman (arXiv:1612.08085) which in part is addressed to the same topic.