PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 109–117 DOI: 10.2298/PIM150126018M

INDEPENDENCE COMPLEXES OF COMAXIMAL GRAPHS OF COMMUTATIVE RINGS WITH IDENTITY

Nela Milošević

Abstract. We study topology of the independence complexes of comaximal (hyper)graphs of commutative rings with identity. We show that the independence complex of comaximal hypergraph is contractible or homotopy equivalent to a sphere, and that the independence complex of comaximal graph is almost always contractible.

1. Introduction

To any graph or hypergraph G we can associate its independence complex, which is the abstract simplicial complex formed from all the independent sets of G, and then study topology of its geometric realization. In this paper we investigate independence complexes of the comaximal graph and the comaximal hypergraph associated with a commutative ring with identity and determine their homotopy type. The set of vertices of the comaximal graph are the elements of the ring R, and two distinct vertices x and y are adjacent if and only if Rx + Ry = R. This graph is denoted by Γ(R) and was first defined by Sharma and Bhatwadekar in [11], who proved that the chromatic number is finite if and only if the ring itself is finite. The graph was further investigated by Maimani et al. in [6] (who actually coined the name comaximal) and by Moconja and Petrović in [8] completing the structure of the comaximal graph in case the number of maximal ideals in R is finite. Similarly, we define comaximal hypergraph H(R) as a generalization of the comaximal graph: vertices are the elements of the ring R and hyperedges (or simply edges) are nonempty subsets of R which are generating sets for the ring, that is {x_{0}, x_{1}, …, x_{n}} is a hyperedge if and only if Rx_{0} + ⋯+ Rx_{n} = R.

In general, the idea of associating a combinatorial object to a commutative ring with identity has been of great interest to researches, so for other examples of associating a graph to a commutative ring the reader may wish to consult [1, 2, 3, 10, 13]. For another example of associating simplicial complexes to commutative rings, we refer the reader to [7] where the authors associated order complex with a general commutative ring via chains of ideals following a suggestion from Vassiliev in [12], and determined homotopy type of that complex.

2010 Mathematics Subject Classification: 55U10; 57M15; 55P15; 05E45.

Key words and phrases: independence complex; graph; simplicial complex; homotopy type; commutative ring.

Communicated by Zoran Petrović.

2. Preliminaries

In this section we collect some definitions mainly concerning simplicial complexes and independence complexes which are needed for the discussion in Section 3 and the exposition of our main results. For more thorough background we refer the reader to [5, 9].

Definition 2.1. An abstract simplicial complex K is a set A together with a collection K of finite nonempty subsets of A such that if X ∈ K and Y ⊆ X, then Y ∈ K.

The element v ∈ A such that {v} ∈ K is called a vertex and the set of all vertices is denoted by V (K). The elements of K are called simplices and usually denoted by σ. The dimension of a simplex σ is |σ| − 1, where |σ| is the number of elements in the set σ. Any nonempty subset of a simplex is called a face of that simplex; faces are simplices themselves. Those simplices that are not faces of any other simplex in K are called maximal. For a simplex τ that is contained in only one maximal simplex σ in the complex, we say that τ is a free face of σ.

Definition 2.2. Let K and L be two abstract simplicial complexes. A simplicial map from K to L is a set map f: V (K) → V (L) such that if {x_{0}, …, x_{n}} is a simplex of K, then {f(x_{0}), …, f(x_{n})} is a simplex of L. We simply write f: K → L. Furthermore, simplicial map f is an isomorphism of abstract simplicial complexes if the induced map is a bijection and its inverse is a simplicial map as well.

Now we turn to the definitions regarding geometric realization.

Definition 2.3. A geometric n-simplex σ is the convex hull of the set A of n+1 affine independent points in ℝ^{N} for some N > n. The standard n-simplex Δ^{n} is the convex hull of the set of the endpoints of the standard unit basis in ℝ^{n+1} .

There is an affine bijection between any geometric n-simplex and the standard geometric n-simplex.

Let us denote by ℝ^{⊕J} direct sum of |J| (where J may be infinite; |J| stands here for the cardinality of J) copies of ℝ (so, it is a subset of ℝ^{J} consisting of those x = (x_{j})_{j∈J} ∈ ℝ^{J} such that x_{j} = 0, for all but finitely many j ∈ J). A geometric simplicial complex K in ℝ^{⊕J} is a collection of geometric simplices in ℝ^{⊕J} such that every face of a simplex in K is a simplex of K and the intersection of any two simplices of K is a face of each of them.

For a geometric simplicial complex K, let |K| denote the union of all simplices of K. We can define a topology on |K|: every simplex σ of K has the induced topology, and a subset F of |K| is closed if and only if F ∩ σ is closed for every simplex σ ∈ K. Topological space |K|, called geometric realization of K, is determined up to a homeomorphism. When we refer to the topological properties of the abstract simplicial complex K, we are always actually referring to the topological space |K|.

Note that a simplicial map f: K → L induces a continuous map |f|: |K| → |L| of topological spaces.

Definition 2.4. A simplicial complex K on a vertex set V (K) is a cone with apex v ∈ V (K) if every σ ∈ K satisfies σ ∪ {v} ∈ K.

If a simplicial complex K is a cone with apex v then its geometric realization can be contracted to a point.

We will also need the following lemma from [9].

Lemma 2.1. [9, Lemma 2.5] If K is finite, then |K| is compact. Conversely, if a subset A of |K| is compact, then A ⊂ |K_{0}| for some finite subcomplex K_{0} of K.

Note that this lemma is formulated for simplicial complex K that lies in ℝ^{N} for some N, which puts limitation on the cardinality of K and on the dimension of the simplices of K. However, the author in [9] removes these restrictions later in the book, and asserts that this result holds in general, which we will use.

There is a large variety of complexes whose description is purely combinatorial, one of which are independence complexes. Given any (hyper)graph G, a set of vertices S ⊆ V (G) is called independent if it contains no (hyper)edge (as its subset).

Definition 2.5. For an arbitrary graph (hyper)G, the independence complex of G, called Ind_{G}, is the abstract simplicial complex whose set of vertices is V (G) and whose simplices are all finite independent sets of G.

In [4] the authors focus on the homotopic properties of a simplicial complex K in terms of its nonfaces, that is, the family {A ⊆ V (K): A ∉ K}. A minimal nonface of a simplicial complex is called a circuit.

Definition 2.6. A simplicial complex K on the vertex set V is constrictive if the complex K is the boundary of the simplex on the vertex set V or there is a vertex v in V belonging to at most one circuit with one of the following properties:

• v belongs to no circuit; or

• v belongs to a unique circuit B ≠ V and there is a vertex u ∉ B such that contracting the edge {u, v} yields a constrictive complex, where an edge {u, v} is contractible if and only if no circuit contains {u, v}.

A constrictive complex K is simple-homotopic to a single vertex or to the boundary complex of a simplex.

3. Independence complex of comaximal graph and comaximal hypergraph

Simplices in the independence complex of the comaximal hypergraph Ind_{H(R)} are independent sets of H(R), that is nongenerating sets in R, {x_{0}, x_{1}, …, x_{n}} ∈ Ind_{H(R)} if and only if Rx_{0} + ⋯+ Rx_{n} ≠ R. This naturally forms a simplicial complex since if an (n+1)-tuple does not generate the entire ring, then any smaller subset certainly does not generate the entire ring. Simplices in the the independence complex of comaximal graph Ind_{Γ(R)} are the independent sets of Γ(R), that is simplices are the sets {x_{0}, x_{1}, …, x_{n}} such that Rx_{i} + Rx_{j} ≠ R for all the pairs of distinct indices 0 ⩽ i, j ⩽ n.

Whenever Rx_{0} + ⋯+ Rx_{n} ≠ R, then {x_{i}, x_{j} } is not an edge in Γ(R) for any 0 ⩽ i, j ⩽ n and i ≠ j, so {x_{0}, x_{1}, …, x_{n}} is an independent set in Γ(R) and hence a simplex in Ind_{Γ(R)} . Therefore the independence complex of comaximal hypergraph is a subcomplex of the independence complex of comaximal graph. We will then first look at the independence complex of comaximal hypergraph and determine homotopy type of its geometric realization and then use these results and extend them to the investigation of independence complex of comaximal graph.

Note that in both complexes units are not adjacent to any vertex and elements of J(R) are adjacent to every nonunit. Therefore, it will be more interesting to exclude units and elements of Jacobson radical from the set of vertices in these complexes; if we were to include elements of J(R), then we would always have contractible complexes whenever J(R) ≠ 0. Thus, we will be looking at the independence complexes of the subgraph of the comaximal hypergraph H′(R) and the subgraph of the comaximal graph Γ′_{2}(R) whose vertex set is V (H′(R)) = V (Γ′_{2}(R)) = R∖(U(R)∪J(R)) (we use same notation for subgraph Γ′_{2}(R) as in [6]). We denote these complexes by Ind_{H′(R)} and Ind_{Γ′_{2} (R)} respectively, and determine their homotopy type.

Remark 3.1. Studying topology of independence complexes has been of interest to researchers and in [4] Ehrenborg and Hetyei studied topology of a certain class of independence complexes called constrictive complexes. They showed that constrictive complexes are either contractible or homotopy equivalent to a sphere. Conclusions concerning the independence complexes of the comaximal graph and comaximal hypergraph are similar to those regarding constrictive complexes, so it is natural to question whether the results can be derived by proving that the complexes we are studying are constrictive. It turns out that the independence complexes we are studying are not always constrictive, which we show by providing a counterexample. Let R = ℤ/p_{1}^{2}p_{2}ℤ where p_{1} and p_{2} are distinct prime numbers, and consider the complexes Ind_{H′(R)} and Ind_{Γ′_{2} (R)} which are equal for this ring. Let a be any vertex in this complex. Since elements of J(R) are not in the vertex set, a is divisible by either p_{1} or p_{2}, but not by both. The geometric realization of this complex consists of two disconnected cones with vertices being elements of each maximal ideal that are not in J(R). Clearly this complex is not a boundary of a simplex, and since each cone has at least two vertices, then every vertex of the complex belongs to more than one circuit. Therefore this complex is not constrictive.

We will determine homotopy type of the independence complex of comaximal hypergraph Ind_{H′(R)} and the independence complex of comaximal graph Ind_{Γ′_{2} (R)} using a direct approach by considering three cases: (1) local rings, (2) rings with infinitely many maximal ideals, and (3) semilocal rings.

Observe that in the case when the ring R is local with the maximal ideal M, the resulting complexes Ind_{H′(R)} and Ind_{Γ′_{2} (R)} are empty since every nonunit in R is contained in M (we have J(R) = M). Also, note that the equivalent definition of Ind_{H′(R)} is {x_{0}, x_{1}, …, x_{n}} ∈ Ind_{H′(R)} if and only if there exists a maximal ideal M in R such that x_{0}, x_{1}, …, x_{n} ∈ M.

3.1. Independence complexes for rings with infinitely many maximal ideals. Consider the case when the ring R has infinitely many maximal ideals. To determine the homotopy type of the independence complex Ind_{H′(R)} in this case, first we need the following lemma.

Lemma 3.1. Suppose that R is such that the set of maximal ideals Max(R) is infinite. If K_{0} is a finite subcomplex of Ind_{H′(R)} , then there is a subcomplex K_{1} of Ind_{H′(R)} such that K_{0} is a subcomplex of K_{1} and |K_{1}| is contractible.

Proof. Let σ_{1}, …, σ_{m} be all the maximal simplices in K_{0}. For every simplex σ_{i} = {x_{i0}, …, x_{in_{i}} } ∈ K_{0} there exists a maximal ideal M_{i} containing elements x_{i0}, …, x_{in_{i}} , so we have a finite set of maximal ideals {M_{1}, …, M_{m}} (not necessarily distinct) containing elements which are vertices of appropriate maximal simplices. We will show that there exists an element y ∈ R ∖ (U(R) ∪ J(R)) such that {x_{i0}, …, x_{in_{i}} , y} ∈ Ind_{H′(R)} for all 1 ⩽ i ⩽ m. This will show that | Ind_{H′(R)} | is connected.

Let y be an element of ⋂ _{i=1}^{m} M_{i}, such that y ∉ J(R). Such element exists because the intersection of finitely many maximal ideals is not equal to J(R). Namely if we were to have ⋂ _{i=1}^{m} M_{i} = J(R), then for some other maximal ideal M_{m+1} ∉ {M_{1}, …, M_{m}} we would have ⋂ _{i=1}^{m} M_{i} ⊂ M_{m+1} , which would mean that M_{i} ⊆ M_{m+1} for some i ∈ {1, …, m} which is a contradiction.

Consider K_{1} as a subcomplex of Ind_{H′(R)} spanned by all of the vertices of K_{0} together with the vertex y (note that y might already be a vertex of K_{0}, in that case K_{1} = K_{0}). For every simplex σ_{i} = {x_{i0}, …, x_{in_{i}} } ∈ K_{0}, we have x_{i0}, …, x_{in_{i}} , y ∈ M_{i}, hence σ_{i} is a face of {x_{i0}, …, x_{in_{i}} , y} ∈ K_{1} so |K_{1}| is a cone with the apex y, hence contractible. □

We have a similar lemma for the independence complex of comaximal graph Ind_{Γ′_{2} (R)} .

Lemma 3.2. Suppose that R is such that the set of maximal ideals Max(R) is infinite. If K_{0} is a finite subcomplex of Ind_{Γ′_{2} (R)} , then there is a subcomplex K_{1} of Ind_{Γ′_{2} (R)} such that K_{0} is a subcomplex of K_{1} and |K_{1}| is contractible.

Proof. Let σ_{1}, …, σ_{m} be all the maximal simplices in K_{0}. For every simplex σ_{i} = {x_{i0}, …, x_{in_{i}} } ∈ K_{0} and every pair {x_{ij} , x_{ik} }, j ≠ k, that is a face of σ_{i}, there exists a maximal ideal M containing elements x_{ij} , x_{ik}, so we have some finite set of maximal ideals {M_{1}, …, M_{t}} containing pairs of elements which are vertices of appropriate maximal simplices. We will show that there exists an element y ∈ R ∖ (U(R) ∪ J(R)) such that {x_{i0}, …, x_{in_{i}} , y} ∈ Ind_{Γ′_{2} (R)} for all 1 ⩽ i ⩽ m.

This will show that | Ind_{Γ′_{2} (R)} | is connected. Let y be an element of ⋂ _{i=1}^{t} M_{i}, such that y ∉ J(R). As in the previous lemma, such element always exists. Consider K_{1} as a subcomplex of Ind_{Γ′_{2} (R)} spanned by all of the vertices of K_{0} together with the vertex y (note that y might already be a vertex of K_{0}, in that case K_{1} = K_{0}). For every simplex σ_{i} = {x_{i0}, …, x_{in_{i}} } ∈ K_{0}, we have {x_{ij} , y} in some maximal ideal for every 0 ⩽ j ⩽ n_{i}, hence {x_{ij} , y} is not an edge in Γ′_{2}(R). Therefore {x_{i0}, …, x_{in_{i}} , y} is an independent set and therefore a simplex in the complex Ind_{Γ′_{2} (R)} . We have that each σ_{i} is a face of {x_{i0}, …, x_{in_{i}} , y} ∈ K_{1} so |K_{1}| is a cone with the apex y, hence contractible. □

Theorem 3.1. If Max(R) is infinite, then | Ind_{H′(R)} | and | Ind_{Γ′_{2} (R)} | are contractible.

Proof. The proof is completely analogous to the proof in [7] which we give here for the sake of completeness. It applies for both complexes so let Ind stand for both Ind_{H′(R)} and Ind_{Γ′_{2} (R)} .

Since | Ind| has the homotopy type of a CW complex, we may use the Whitehead theorem. We only need to show that all homotopy groups of | Ind| are trivial. Suppose that n ⩾ 1 and that g: S^{n} → | Ind| is a continuous map. Since the image g[S^{n}] is compact, by Lemma 2.1 there is a finite subcomplex K_{0} such that g[S^{n}] ⊆ |K_{0}|. By Lemma 3.1, there is a subcomplex K_{1} such that K_{0} ⊂ K_{1} and |K_{1}| is contractible. So, the map g may be factored through the contractible space |K_{1}| and it is homotopically trivial. We conclude that π_{n}(| Ind|, ∗) is trivial. Since this holds for all n, by Whitehead’s theorem we get that | Ind| is contractible. □

3.2. Independence complex of comaximal hypergraph when R is a semilocal ring. Let R be a semilocal ring with m maximal ideals M_{1}, M_{2}, …, M_{m}. We can easily observe that the maximal simplices in Ind_{H′(R)} are formed from the set of elements in each maximal ideal. Therefore it will be useful to first look at a related complex whose vertices are proper ideals of the ring R (naturally excluding J(R)), namely the nongenerating ideal complex.

Definition 3.1. Let R be a commutative ring and let I^{∗}(R) be the set of all proper ideals of R properly containing J(R). The nongenerating ideal complex 𝒞 (R) is defined by:

V (𝒞 (R)) = I^{∗}(R), and {I_{0}, …, I_{n}} ∈ 𝒞 (R) if and only if I_{0} + ⋯+ I_{n} ≠ R

Again, we have an equivalent definition {I_{0}, …, I_{n}} ∈ 𝒞 (R) if and only if I_{0}, …, I_{n} ⊆ M for some maximal ideal M. Consider the subcomplex K̃ of 𝒞 (R) whose vertices are all the maximal ideals and their intersections (except the intersection of all of them), that is, vertices are ⋂ _{i∈S} M_{i} where S is any proper nonempty subset of the set {1, 2, …, m}. Geometric realization of this subcomplex for the case m = 3 is given below in Figure 1.

Note that this complex is connected unless | Max(R)| = 2.

It will be useful to denote [m] = {1, 2, …, m}, while for the intersection (union) of all maximal ideals whose indices belong to a set S ⊂ [m] we use the same notation as in [8], that is M_{S} = ⋂ _{i∈S} M_{i} and M^{S} = ⋃ _{i∈S} M_{i}.

\begin{matrix}M_{1} ∩ M_{2} \\ M_{1} ∩ M_{3} \\ M_{2} ∩ M_{3} \\ M_{1} \\ M_{2} \\ M_{3} \end{matrix} ; Figure 1.

Lemma 3.3. Let R be a semilocal ring. If | Max(R)| = m > 1, then |K̃| ≃ Δ̇^{m−1} .

Proof. Note that there are exactly m maximal simplices in K̃, namely simplices whose vertices are ideals contained in M_{i} for each i = 1, …, m. For each such maximal simplex σ_{i}, consider its face τ_{i} whose vertices are ideals M_{[m]∖{j}} for each j = 1, …, m and j ≠ i. Let K_{0} be the subcomplex with m vertices M_{[m]∖{j}} for each j = 1, …, m. Then K_{0} has exactly m maximal simplices τ_{1}, …, τ_{m}, and its geometric realization is the boundary of an (m−1)-simplex. We define a continuous function f: |K̃| → |K_{0}| by mapping each vertex M_{S} in |K̃| to the barycenter of the simplex whose vertices are M_{[m]∖{j}} for each j = 1, …, m and j ∉ S (this is exactly the simplex ⋂ _{i∈S} τ_{i}), and any point x = ∑ _{i=0}^{n} a_{i}M_{S_{i}} to f(x) = ∑ _{i=0}^{n} a_{i}f(M_{S_{i}} ). In this manner, each maximal simplex σ_{i} is projected onto its face τ_{i}, as any point of |K̃| which is inside σ_{i} is projected onto appropriate point of τ_{i} along the line that is inside the simplex σ_{i}. Then f is a strong deformation retraction, so |K̃| ≃ |K_{0}|, that is, |K̃| ≃ Δ̇^{m−1} . □

For example, in case |Max(R)| = 3 (see Figure 1), there are three maximal simplices (in the geometric realization they are represented by three full triangles), which we are projecting onto the appropriate edge of the hollow triangle in the center. The function f maps vertex M_{1} to the barycenter of the simplex {M_{1} ∩ M_{2}, M_{1} ∩ M_{3}}, M_{2} to the barycenter of the simplex {M_{1} ∩ M_{2}, M_{2} ∩ M_{3}}, and M_{3} to the barycenter of the simplex {M_{1} ∩ M_{3}, M_{2} ∩ M_{3}}.

Now we can finally determine the homotopy type of the independence complex Ind_{H′(R)} when R is a semilocal ring.

Theorem 3.2. Let R be a semilocal ring with |Max(R)| = m > 1; then | Ind_{H′(R)} | ≃ Δ̇^{m−1} .

Proof. In order to prove this, we show that | Ind_{H′(R)} | ≃ |K̃| and use this together with the above lemma.

For each proper nonempty subset S ⊂ [m], the set M_{S} ∖ M^{S^{c}} is nonempty by the Prime avoidance lemma. Choose an element a_{S} ∈ M_{S} ∖ M^{S^{c}} , and consider subcomplex \overline{Ind}_{H′(R)} of the independence complex whose vertices are such elements. Consider a simplicial map between \overline{Ind}_{H′(R)} and the complex K̃ discussed above, given by the vertex map f : a_{S} ↦ M_{S}. This map is well defined and a bijection, since a_{S} = a_{T} if and only if S = T, that is, if and only if M_{S} = M_{T}. Furthermore this is an isomorphism because \{a_{S_{0}} , …, a_{S_{n}} \} ∈ \overline{Ind}_{H′(R)} if and only if there exists i ∈ S_{0} ∩ ⋯ ∩ S_{n} if and only if M_{S_{0}} , …, M_{S_{n}} ⊆ M_{i} if and only if {M_{S_{0}} , …, M_{S_{n}} } ∈ K̃. Therefore the complexes \overline{Ind}_{H′(R)} and K̃ are isomorphic, hence |\overline{Ind}_{H′(R)} | ≃ |K̃|.

Let g : | Ind_{H′(R)} | → |\overline{Ind}_{H′(R)} | be the simplicial map such that a vertex x is mapped to a_{S} where S is the set of indices of all maximal ideals containing x, that is, we are mapping each element of R∖(U(R)∪J(R)) to appropriate representative of the intersection of the maximal ideals containing that element. For any simplex {x_{0}, …, x_{n}} in Ind_{H′(R)} there is a maximal ideal containing elements x_{0}, …, x_{n}, which then also contains g(x_{0}), …, g(x_{n}). Hence, simplex {g(x_{0}), …, g(x_{n})} in Ind_{H′(R)} is just a face of the simplex {x_{0}, …, x_{n}, g(x_{0}), …, g(x_{n})} and our map is the projection of that larger simplex onto its face. This is clearly a strong deformation retraction since every point in |Ind_{H′(R)} | is mapped onto its image along a line inside the appropriate simplex. Therefore |Ind_{Γ′_{2} (R)} | ≃ |\overline{Ind}_{H′(R)} | ≃ |K̃|, so |Ind_{H′(R)} | ≃ Δ̇^{m−1} . □

3.3. Independence complex of comaximal graph when R is a semilocal ring. Unlike the independence complex of comaximal hypergraph, we will show that the independence complex of comaximal graph is contractible for semilocal rings with more than two maximal ideals. To show this we use a similar idea as before.

Theorem 3.3. Let R be a semilocal ring with m maximal ideals M_{1}, …, M_{m}. If m = 2, then | Ind_{Γ′_{2} (R)} | is homotopy equivalent to a pair of points with the discrete topology, and if m > 2 the complex | Ind_{Γ′_{2} (R)} | is contractible.

Proof. Elements of maximal ideals form simplices in the complex (because no pair of elements generates entire ring), so again for each proper nonempty subset S ⊂ [m] we choose an element a_{S} ∈ M_{S} ∖ M^{S^{c}} , and consider subcomplex \overline{Ind}_{Γ′_{2} (R)} of the independence complex whose vertices are such elements. Let g: | Ind_{Γ′_{2} (R)} | → |\overline{Ind}_{Γ′_{2} (R)} | be the simplicial map such that a vertex x is mapped to a_{S} where S is the set of indices of all maximal ideals containing x. Then for any simplex {x_{0}, …, x_{n}} in Ind_{Γ′_{2} (R)} we also have {x_{0}, …, x_{n}, g(x_{0}), …, g(x_{n})} in Ind_{Γ′_{2} (R)} because every pair is in some maximal ideal hence not generating entire ring; namely {x_{i}, g(x_{j})} (i and j need not be distinct) is in the same maximal ideal as is {x_{i}, x_{j} }, and {g(x_{i}), g(x_{j})} is in the same maximal ideal as {x_{i}, x_{j} }. Now, same as in the proof of Theorem 3.2, we conclude that g is a strong deformation retraction, so | Ind_{Γ′_{2} (R)} | ≃ |\overline{Ind}_{Γ′_{2} (R)} |. Now consider the map f: |\overline{Ind}_{Γ′_{2} (R)} | → |\overline{Ind}_{Γ′_{2} (R)} | that sends a_{S} to the barycenter of the simplex whose vertices are a_{[m]∖{j}} for each j = 1, …, m and j ∉ S. Every point of each simplex σ is projected onto appropriate point of its face along the line that is inside σ so this is a strong deformation retraction. Therefore the complex |\overline{Ind}_{Γ′_{2} (R)} | is reduced to its subcomplex consisting of the vertices a_{[m]∖{1}} , …, a_{[m]∖{m}} , and for m > 2 these vertices form a simplex because every pair of elements is in some maximal ideal. Therefore for m > 2 the independence complex Ind_{Γ′_{2} (R)} is contractible while for m = 2 it is homotopy equivalent to two disjoint points. □

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Faculty for Information Systems and Technologies (Received 23 10 2014) University of Donja Gorica (Revised 26 01 2015 and 06 08 2015) Podgorica Montenegro

 nela.milosevic@udg.edu.me