Abstract

Let 𝐿 be a complete lattice. We introduce and investigate the 𝐿-total graph of an 𝐿-module over an 𝐿-commutative ring. The main purpose of this paper is to extend the definition and results given in (Anderson and Badawi, 2008) to more generalize the 𝐿-total graph of an 𝐿-module case.

1. Introduction

It was Beck (see [1]) who first introduced the notion of a zero-divisor graph for commutative rings. This notion was later redefined by Anderson and Livingston in [2]. Since then, there has been a lot of interest in this subject, and various papers were published establishing different properties of these graphs as well as relations between graphs of various extensions (see [25]). Let 𝑅 be a commutative ring with 𝑍(𝑅)being its set of zero-divisors elements. The total graph of 𝑅, denoted by 𝑇(Γ(𝑅)), is the (undirected) graph with all elements of 𝑅 as vertices, and, for distinct 𝑥,𝑦𝑅, the vertices 𝑥 and 𝑦 are adjacent if and only if 𝑥+𝑦𝑍(𝑅). The total graph of a commutative ring has been introduced and studied by Anderson and Badawi in [3]. In [6], the notion of the total torsion element graph of a module over a commutative ring is introduced.

In [7], Zadeh introduced the concept of fuzzy set, which is a very useful tool to describe the situation in which the data is imprecise or vague. Many researchers used this concept to generalize some notions of algebra. Goguen in [8] generalized the notion of fuzzy subset of 𝑋 to that of an 𝐿-subset, namely, a function from 𝑋 to a lattice 𝐿. In [9], Rosenfeld considered the fuzzification of algebraic structures. Liu [10] introduced and examined the notion of a fuzzy ideal of a ring. Since then several authors have obtained interesting results on 𝐿-ideals of a ring 𝑅 and 𝐿-modules (see [11, 12]). Also, 𝐿-zero-divisor graph of an 𝐿-commutative ring has been introduced and studied in [13].

In the present paper we introduce a new class of graphs, called the 𝐿-total torsion element graph of a 𝐿-module (see Definition 2.2), and we completely characterize the structure of this graph. The total torsion element graph of a module over a commutative ring and the 𝐿-total torsion element graph of a 𝐿-module over a 𝐿-commutative ring are different concepts. Some of our results are analogous to the results given in [6]. The corresponding results are obtained by modification, and here we give a complete description of the 𝐿-total torsion element graph of an 𝐿-module.

For the sake of completeness, we state some definitions and notation used throughout. For a graph Γ, by 𝐸(Γ) and 𝑉(Γ), we denote the set of all edges and vertices, respectively. We recall that a graph is connected if there exists a path connecting any two distinct vertices. The distance between two distinct vertices 𝑎 and 𝑏, denoted by 𝑑(𝑎,𝑏), is the length of the shortest path connecting them (if such a path does not exist, then 𝑑(𝑎,𝑎)=0 and 𝑑(𝑎,𝑏)=). The diameter of a graph Γ, denoted by diam(Γ), is equal to sup{𝑑(𝑎,𝑏)𝑎,𝑏𝑉(Γ)}. A graph is complete if it is connected with diameter less than or equal to one. The girth of a graph Γ, denoted gr(Γ), is the length of the shortest cycle in Γ, provided Γ contains a cycle; otherwise, gr(Γ)=. We denote the complete graph on 𝑛 vertices by 𝐾𝑛 and the complete bipartite graph on 𝑚 and 𝑛 vertices by 𝐾𝑚,𝑛 (we allow 𝑚 and 𝑛 to be infinite cardinals). We will sometimes call a 𝐾1,𝑚 a star graph. We say that two (induced) subgraphs Γ1 and Γ2 of Γ are disjoint if Γ1 and Γ2 have no common vertices and no vertex of Γ1 (resp., Γ2) is adjacent (in Γ) to any vertex not in Γ1 (resp., Γ2).

Let 𝑅 be a commutative ring, and 𝐿 stands for a complete lattice with least element 0 and greatest element 1. By an 𝐿-subset 𝜇 of a nonempty set 𝑋, we mean a function 𝜇 from 𝑋 to 𝐿. If 𝐿=[0,1], then 𝜇 is called a fuzzy subset of 𝑋. 𝐿𝑋 denotes the set of all 𝐿-subsets of 𝑋. We recall some definitions and lemmas from the book [12], which we need for development of our paper.

Definition 1.1. An 𝐿-ring is a function 𝜇𝑅𝐿, where (𝑅,+,.) is a ring, which satisfies the following.(1)𝜇0.(2)𝜇(𝑥𝑦)𝜇(𝑥)𝜇(𝑦) for every 𝑥,𝑦 in 𝑅.(3)𝜇(𝑥𝑦)𝜇(𝑥)𝜇(𝑦) for every 𝑥,𝑦 in 𝑅.

Definition 1.2. Let 𝜇𝐿𝑅. Then 𝜇 is called an 𝐿-ideal of 𝑅 if for every 𝑥,𝑦𝑅 the following conditions are satisfied. (1)𝜇(𝑥𝑦)𝜇(𝑥)𝜇(𝑦).(2)𝜇(𝑥𝑦)𝜇(𝑥)𝜇(𝑦).
The set of all 𝐿-ideals of 𝑅 is denoted by 𝐿𝐼(𝑅).

Definition 1.3. Assume that 𝑀 is an 𝑅-module, and let 𝜇𝐿𝑀. Then 𝜇 is called an 𝐿-fuzzy 𝑅-module of 𝑀 if for all 𝑥,𝑦𝑀 and for all 𝑟𝑅 the following conditions are satisfied.(1)𝜇(𝑥𝑦)𝜇(𝑥)𝜇(𝑦).(2)𝜇(𝑟𝑥)𝜇(𝑥).(3)𝜇(0𝑀)=𝜇(1).
The set of all 𝐿-fuzzy 𝑅-modules of 𝑀 is denoted by 𝐿(𝑀).

Lemma 1.4. Let 𝑀 be a module over a ring 𝑅, and 𝜇𝐿(𝑀). Then 𝜇(𝑚)𝜇(0𝑀) for every 𝑚𝑀.

2. 𝑇(𝜇) Is a Submodule of 𝑀

Let 𝑀 be a module over a commutative ring 𝑅, and let 𝜇𝐿(𝑀). The structure of the 𝐿-total torsion element graph 𝑇(Γ(𝜇)) may be completely described in those cases when 𝜇-torsion elements form a submodule of 𝑀. We begin with the key definition of this paper.

Definition 2.1. Let 𝑀 be a module over a commutative ring 𝑅, and let 𝜇𝐿(𝑀). A 𝜇-torsion element is an element 𝑚𝑀 with 𝜇(𝑚)𝜇(0𝑀) for which there exists a nonzero element 𝑟 of 𝑅 such that 𝜇(𝑟𝑚)=𝜇(0𝑀).

The set of 𝜇-torsion elements in 𝑀 will be denoted by 𝑇(𝜇).

Definition 2.2. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀). We define the 𝐿-total torsion element graph of an 𝐿-module 𝑇(Γ(𝜇)) as follows: 𝑉(𝑇(Γ(𝜇)))=𝑀, 𝐸(𝑇(Γ(𝜇)))={{𝑥,𝑦}𝑥+𝑦𝑇(𝜇)}.

Notation 1. For the 𝜇-torsion element graph 𝑇(Γ(𝜇)), we denote the diameter, the girth, and the distance between two distinct vertices 𝑎 and 𝑏, by diam(𝑇(Γ(𝜇))), gr(𝑇(Γ(𝜇))), and 𝑑𝜇(𝑎,𝑏), respectively.

Remark 2.3. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀). Clearly, if 𝜇 is a nonzero constant, then 𝑇(Γ(𝜇))=. So throughout this paper, we will assume, unless otherwise stated, that 𝜇 is not a nonzero constant. Thus, there is a nonzero element 𝑦 of 𝑀 such that 𝜇(𝑦)𝜇(0𝑀).

We will use Tof(𝜇) to denote the set of elements of 𝑀 that are not 𝜇-torsion elements. Let Tof(Γ(𝜇)) be the (induced) subgraph of 𝑇(Γ(𝜇)) with vertices Tof(𝜇), and let Tor(Γ(𝜇)) be the (induced) subgraph of 𝑇(Γ(𝜇)) with vertices 𝑇(𝜇).

Definition 2.4. Let 𝑀 be a module over a ring 𝑅, and 𝜇𝐿(𝑀). One defines the set ann𝜇(𝑀) by ann𝜇(𝑀)={𝑟𝑅𝜇(𝑟𝑀)={𝜇(0𝑀)}}, the 𝜇-annihilator of 𝑀.

Lemma 2.5. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀). Then ann𝜇(𝑀) is an 𝐿-ideal of 𝑅.

Proof. Let 𝑟,𝑠ann𝜇(𝑀) and 𝑡𝑅. If 𝑚𝑀, then we have 𝜇((𝑟𝑠)𝑚)𝜇(𝑟𝑚)𝜇(𝑠𝑚)=𝜇(0𝑀)𝜇(0𝑀)=𝜇(0𝑀) and 𝜇(𝑡𝑟𝑚)=𝜇(𝑡(𝑟𝑚))𝜇(𝑟𝑚)=𝜇(0𝑀). It then follows from Lemma 1.4 that 𝜇((𝑟𝑠)𝑚)=𝜇(0𝑀); hence 𝑟𝑠ann𝜇(𝑀). Similarly, 𝑟𝑡ann𝜇(𝑀).

Theorem 2.6. Let 𝑀 be a module over a ring 𝑅 and let 𝜇𝐿(𝑀). Then the 𝐿-torsion element graph 𝑇(Γ(𝜇)) is complete if and only if 𝑇(𝜇)=𝑀.

Proof. If 𝑇(𝜇)=𝑀, then for any vertices 𝑚,𝑚𝑀, one has 𝑚+𝑚𝑇(𝜇); hence they are adjacent in 𝑇(Γ(𝜇)). On the other hand, if 𝑇(Γ(𝜇)) is complete, then every vertex is adjacent to 0. Thus,𝑚=𝑚+0𝑇(𝜇) for every 𝑚𝑀. This completes the proof.

Theorem 2.7. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a submodule of 𝑀. Then one has the following.(i)Tor(Γ(𝜇)) is a complete (induced) subgraph of 𝑇(Γ(𝜇)) and Tor(Γ(𝜇)) is disjoint from Tof(Γ(𝜇)).(ii)If annμ(𝑀)0, then 𝑇(Γ(𝜇)) is a complete graph.

Proof. (i) Tor(Γ(𝜇)) is complete directly from the definition. Finally, if 𝑚𝑇(𝜇) and 𝑚Tof(𝜇) were adjacent, then 𝑚+𝑚𝑇(𝜇); so this, since 𝑇(𝜇) is a submodule, would lead to the contradiction 𝑚𝑇(𝜇).
(ii) Let 𝑚𝑀. we may assume that 𝜇(𝑚)𝜇(0𝑀). By assumption, there exists 0𝑠𝑅 with 𝜇(𝑠𝑀)=𝜇(0𝑀), so 𝜇(𝑠𝑚)=𝜇(0𝑀). Thus 𝑚𝑇(𝜇), and; therefore, 𝑇(Γ(𝜇)) is a complete graph by Theorem 2.6.

Theorem 2.8. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀). Then 𝑇(Γ(𝜇)) is totally disconnected if and only if 𝑅 has characteristic 2 and 𝑇(𝜇)={0𝑀}.

Proof. If 𝑇(𝜇)={0𝑀}, then the vertices 𝑚1 and 𝑚2 are adjacent if and only if 𝑚1=𝑚2. Then 𝑇(Γ(𝜇)) is a disconnected graph, and its only edges are those that connect vertices 𝑚𝑖 and 𝑚𝑖 (we do not need a priori assumption that 𝑅 has characteristic 2). Conversely, assume that 𝑇(Γ(𝜇)) is totally disconnected. Then 0+𝑚𝑇(𝜇) for every nonzero element 𝑚 of 𝑀. Thus, 𝑇(𝜇)={0𝑀}. Further, since 𝑚+(𝑚)=0, we have 𝑚=𝑚 (so 𝜇(2𝑚)=𝜇(0𝑀)) for every 𝑚𝑀 with 𝜇(𝑚)𝜇(0𝑀) by the total disconnectedness of the graph 𝑇(Γ(𝜇)). As 𝑇(𝜇)={0𝑀}, it follows that 2=1𝑅+1𝑅=0. Thus, char(𝑅)=2.

Proposition 2.9. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a submodule of 𝑀. If 𝑚Tof(𝜇), then 2𝑚𝑇(𝜇) if and only if 2𝑍(𝑅).

Proof. First suppose that 2𝑚𝑇(𝜇). Since 𝑚𝑇(𝜇), we get that 𝜇(𝑚)𝜇(0𝑀), and, for all 𝑟𝑅,𝜇(𝑟𝑚)=𝜇(0𝑀) implies that 𝑟=0. Since 2𝑚𝑇(𝜇), there is a nonzero element 𝑐𝑅 such that 𝜇(𝑐(2𝑚))=𝜇((2𝑐)𝑚)=𝜇(0𝑀), and, since 𝑚𝑇(𝜇), one must have 2𝑐=0; hence, 2𝑍(𝑅). Conversely, assume that 2𝑍(𝑅). Then there exists 0𝑑𝑅 with 2𝑑=0. Since 𝜇(0𝑀)=𝜇((2𝑑)𝑚)=𝜇(𝑑(2𝑚)), we have 2𝑚𝑇(𝜇).

Theorem 2.10. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a proper submodule of 𝑀. Then 𝑇(Γ(𝜇)) is disconnected.

Proof. If 𝑇(𝜇)={0𝑀}, then 𝑇(Γ(𝜇)) is disconnected by Theorem 2.8. If 𝑇(𝜇){0𝑀}, then the subgraphs of Tor(Γ(𝜇)) and Tof(Γ(𝜇)) are disjoint by Theorem 2.7 (i), as required.

Theorem 2.11. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a proper submodule of M. Suppose |𝑇(𝜇)|=𝛼 and |𝑀/𝑇(𝜇)|=𝛽. Then one has the following.(i)If 2𝑍(𝑅), then 𝑇(Γ(𝜇)) is a union of 𝛽 disjoint complete graphs 𝐾𝛼.(ii)If 2𝑍(𝑅), then 𝑇(Γ(𝜇)) is a union of (𝛽1)/2 disjoint bipartite graphs 𝐾𝛼,𝛼 and one complete graph 𝐾𝛼.

Proof. (i) Assume that 2𝑍(𝑅) and let 𝑚,𝑚Tof(𝜇) be such that 𝑚+𝑇(𝜇)𝑚+𝑇(𝜇). The elements 𝑚+𝑡, 𝑚+𝑡 from the same coset 𝑚+𝑇(𝜇) are adjacent if and only if 2𝑚𝑇(𝜇), so 2𝑍(𝑅), according to the Proposition 2.9. Then 𝑚+𝑡 and 𝑚+𝑡 are not adjacent (otherwise, we would have 𝑚𝑚=𝑚+𝑚2𝑚𝑇(𝜇)), and; therefore, 𝑚+𝑇(𝜇)=𝑚+𝑇(𝜇). Since every coset has cardinality 𝛼, we conclude that 𝑇(Γ(𝜇)) is the disjoint union of 𝛽 complete graph 𝐾𝛼.
(ii) If 2𝑍(𝑅), then the elements 𝑚+𝑡, 𝑚+𝑡 from 𝑚+𝑇(𝜇) are obviously not adjacent. The elements 𝑚+𝑡, 𝑚+𝑡 from different cosets are adjacent if and only if 𝑚+𝑚𝑇(𝜇) or 𝑚+𝑇(𝜇)=(𝑚)+𝑇(𝜇). In this way we obtain that the subgraph spanned by the vertices from Tof(𝜇) is a disjoint union of (𝛽1)/2 (=𝛽 if 𝛽 is infinite) disjoint bipartite graph 𝐾𝛼,𝛼.

Proposition 2.12. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a proper submodule of 𝑀. Then one has the following.(i)Tof(Γ(𝜇)) is complete if and only if either |𝑀/𝑇(𝜇)|=2 or |𝑀/𝑇(𝜇)|=|𝑀|=3.(ii)Tof(Γ(𝜇)) is connected if and only if either |𝑀/𝑇(𝜇)|=2 or |𝑀/𝑇(𝜇)|=3.(iii)Tof(Γ(𝜇)) and, hence; (Tor(Γ(𝜇)) and 𝑇(Γ(𝜇))) is totally disconnected if and only if 𝑇(𝜇)={0𝑀} and 2𝑍(𝑅).

Proof. Let |𝑀/𝑇(𝜇)|=𝛽 and |𝑇(𝜇)|=𝛼.(i)Let Tof(Γ(𝜇)) be complete. Then, by Theorem 2.11, Tof(Γ(𝜇)) is complete if and only if Tof(Γ(𝜇)) is a single 𝐾𝛼 or 𝐾1,1. If 2𝑍(𝑅), then 𝛽1=1. Thus, 𝛽=2, and hence |𝑀/𝑇(𝜇)|=2. If 2𝑍(𝑅), then 𝛼=1 and (𝛽1)/2=1. Thus, 𝑇(𝜇)={0} and 𝛽=3; hence, |𝑀|=|𝑀/𝑇(𝜇)|=3. The reverse implication may be proved in a similar way as in [6, Theorem  2.6 (1)].(ii)By theorem 2.11, Tof(Γ(𝜇)) is connected if and only if Tof(Γ(𝜇)) is a single 𝐾𝛼 or 𝐾𝛼,𝛼. Thus, either 𝛽1=1 if 2𝑍(𝑅) or (𝛽1)/2=1 if 2𝑍(𝑅); hence, 𝛽=2 or 𝛽=3, respectively, as needed. The reverse implication may be proved in a similar way as in [3, Theorem  2.6 (2)].(iii)Tof(Γ(𝜇)) is totally disconnected if and only if it is a disjoint union of 𝐾1’s. So by Theorem 2.11, |𝑇(𝜇)|=1 and |𝑀/𝑇(𝜇)|=1, and the proof is complete.

By the proof of the Proposition 2.12, the next theorem gives a more explicit description of the diameter of Tof(Γ(𝜇)).

Theorem 2.13. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a proper submodule of 𝑀. Then one has the following.(i)diam(Tof(Γ(𝜇)))=0 if and only if 𝑇(𝜇)={0} and |𝑀|=2.(ii)diam(Tof(Γ(𝜇)))=1 if and only if either 𝑇(𝜇){0𝑀} and |𝑀/𝑇(𝜇)|=2 or 𝑇(𝜇)={0} and |𝑀|=3.(iii)diam(Tof(Γ(𝜇)))=2 if and only if 𝑇(𝜇){0𝑀} and |𝑀/𝑇(𝜇)|=3.(iv)Otherwise, diam(Tof(Γ(𝜇)))=.

Proposition 2.14. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a proper submodule of 𝑀. Then gr(Tof(Γ(𝜇)))=3,4 or . In particular, gr(Tof(Γ(𝜇)))4 if Tof(Γ(𝜇)) contains a cycle.

Proof. Let Tof(Γ(𝜇)) contain a cycle. Then since Tof(Γ(𝜇)) is disjoint union of either complete or complete bipartite graphs by Theorem 2.11, it must contain either a 3 cycles or a 4 cycles. Thus gr(Tof(Γ(𝜇)))4.

Theorem 2.15. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is a proper submodule of 𝑀. Then one has the following.(i)(a)gr(Tof(Γ(𝜇)))=3 if and only if 2𝑍(𝑅) and |𝑇(𝜇)|3.(b)gr(Tof(Γ(𝜇)))=4 if and only if 2𝑍(𝑅) and |𝑇(𝜇)|2.(c)Otherwise, gr(Tof(Γ(𝜇)))=.(ii)(a)gr(𝑇(Γ(𝜇)))=3 if and only if |𝑇(𝜇)|3.(b)gr(𝑇(Γ(𝜇)))=4 if and only if 2𝑍(𝑅) and |𝑇(𝜇)|=2.(c)Otherwise, gr(𝑇(Γ(𝜇)))=.

Proof. Apply Theorem 2.11, Proposition 2.14, and Theorem 2.7 (i).

The previous theorems give a complete description of the structure of the 𝐿-total torsion element graph of an 𝐿-module 𝑀 when 𝑇(𝜇) is a submodule. The question under what conditions 𝑇(𝜇) is a submodule of 𝑀 and how is this related to the condition that 𝑍(𝑅) is an ideal in 𝑅 naturally arises. We prove that the following results holds.

Theorem 2.16. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀). Then one has the following.(i)If 𝑍(𝑅)={0𝑅}, then 𝑇(𝜇) is a submodule of 𝑀.(ii)If 𝑍(𝑅)=𝑅𝑐 is a principal ideal of 𝑅 with 𝑐 a nilpotent element of 𝑅, then 𝑇(𝜇) is a submodule of 𝑀.

Proof. (i) Let 𝑚,𝑚𝑇(𝜇) and 𝑟𝑅. There are nonzero elements 𝑎,𝑏𝑅 such that 𝜇(𝑚)𝜇(0𝑀), 𝜇(𝑚)𝜇(0𝑀), and 𝜇(𝑎𝑚)=𝜇(𝑏𝑚)=𝜇(0𝑀) with 𝑎𝑏0 (since 𝑅 is an integral domain). It follows that 𝜇(𝑎𝑏(𝑚+𝑚))𝜇(𝑎𝑏𝑚)𝜇(𝑎𝑏𝑚)=𝜇(0𝑀)𝜇(0𝑀)=𝜇(0𝑀); hence, 𝜇(𝑎𝑏(𝑚+𝑚))=𝜇(0𝑀) by Lemma 1.4. Thus, 𝑚+𝑚𝑇(𝜇). Similarly, 𝑟𝑚𝑇(𝜇), and this completes the proof.
(ii) Assume that 𝑇(𝜇) is not a submodule of 𝑀. Then there are elements 𝑚,𝑚𝑇(𝜇) such that 𝑚+𝑚𝑇(𝜇). By assumption, there exist nonzero elements 𝑟,𝑠𝑅 such that 𝜇(𝑟𝑚)=𝜇(0𝑀)=𝜇(𝑠𝑚)=𝜇(0𝑀), where 𝜇(𝑚)𝜇(0𝑀) and 𝜇(𝑚)𝜇(0𝑀). Then 𝜇(𝑟𝑠(𝑚+𝑚))=𝜇(0𝑀) and 𝑚+𝑚𝑇(𝜇), so we must have 𝑟𝑠=0, and; thus, 𝑟,𝑠𝑍(𝑅). Since 𝑐 is nilpotent, we have 𝑟=𝑟1𝑐𝑡 and 𝑠=𝑠1𝑐𝑢, for some 𝑟1,𝑠1𝑍(𝑅). We may assume that 𝑡𝑢. Then for the nonzero element 𝑠1𝑟 of 𝑅 we have 𝜇(𝑠1𝑟(𝑚+𝑚))=𝜇(0𝑀) which is contrary to the assumption that 𝑚+𝑚𝑇(𝜇).

Example 2.17. Assume that 𝑅= is the ring integers, and let 𝑀=𝑅. We define the mapping 𝜇𝑀[0,1] by 1𝜇(𝑚)=2if1𝑥2,5otherwise.(2.1) Then 𝜇𝐿(𝑀) and 𝑇(𝜇)=𝑀. Thus, 𝑇(Γ(𝜇)) is a complete graph by Theorem 2.6.

Example 2.18. Let 𝑀1=𝑅1=𝑍8 denote the ring of integers modulo 8 and 𝑀2=𝑅2=𝑍25 the ring of integers modulo 25. We define the mappings 𝜇1𝑀1[0,1] by 𝜇11(𝑥)=if𝑥=10,2otherwise(2.2) and 𝜇2𝑀2[0,1] by 𝜇2(1𝑚)=if𝑥=10,3otherwise.(2.3) Then, for each 𝑖 (1𝑖2), 𝜇𝑖𝐿(𝑀𝑖), 𝑇(𝜇1)={0,2,4,6}, and 𝑇(𝜇2)={0,5,10,15,20}. An inspection will show that 𝑇(𝜇1) and 𝑇(𝜇2) are submodules of 𝑀1 and 𝑀2, respectively. Therefore, by Theorem 2.11, we have the following results.(1)Since 2𝑍(𝑅1), we conclude that 𝑇(Γ(𝜇1)) is a union of 2 disjoint 𝐾4.(2)Since 2𝑍(𝑅2), we conclude that 𝑇(Γ(𝜇2)) is a disjoint union of 2 complete graph 𝐾5 and 5 bipartite 𝐾5,5.

3. 𝑇(𝜇) Is Not a Submodule of 𝑀

We continue to use the notation already established, so 𝑀 is a module over a commutative ring 𝑅 and 𝜇𝐿(𝑀). In this section, we study the 𝐿-torsion element graph 𝑇(Γ(𝜇)) when 𝑇(𝜇) is not a submodule of 𝑀.

Lemma 3.1. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is not a submodule of 𝑀. Then there are distinct 𝑚,𝑚𝑇(𝜇) such that 𝑚+𝑚Tof(𝜇).

Proof. It suffices to show that 𝑇(𝜇) is always closed under scalar multiplication of its elements by elements of 𝑅. Let 𝑚𝑇(𝜇) and 𝑟𝑅. There is a nonzero element 𝑠𝑅 with 𝜇(𝑠𝑚)=𝜇(0𝑀) such that 𝜇(𝑚)𝜇(0𝑀), so 𝜇(𝑠(𝑟𝑚))=𝜇(𝑟(𝑠𝑚))𝜇(𝑠𝑚)=𝜇(0𝑀); hence, 𝜇(𝑠(𝑟𝑚))=𝜇(0𝑀) by Lemma 1.4, as required.

Theorem 3.2. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is not a submodule of 𝑀. Then one has the following.(i)Tor(Γ(𝜇)) is connected with diam(Tor(Γ(𝜇)))=2.(ii)Some vertex of Tor(Γ(𝜇)) is adjacent to a vertex of Tof(Γ(𝜇)). In particular, the subgraphs Tor(Γ(𝜇)) and Tof(Γ(𝜇)) of 𝑇(Γ(𝜇)) are not disjoint.(iii)If Tof(Γ(𝜇)) is connected, then 𝑇(Γ(𝜇)) is connected.

Proof. (i) Let 𝑥𝑇(𝜇). Then 𝑥 is adjacent to 0. Thus, 𝑥0𝑦 is a path in Tor(Γ(𝜇)) of length two between any two distinct 𝑥,𝑦𝑇(𝜇). Moreover, there exist nonadjacent 𝑥,𝑦𝑇(𝜇) by Lemma 3.1; thus, diam(Tor(Γ(𝜇)))=2.
(ii) By Lemma 3.1, there exist distinct 𝑥,𝑦𝑇(𝜇) such that 𝑥+𝑦Tof(𝜇). Then 𝑥𝑇(𝜇) and 𝑥+𝑦Tof(𝜇) are adjacent vertices in 𝑇(Γ(𝜇)) since 𝑥+(𝑥+𝑦)=𝑦𝑇(𝜇). Finally, the “in particular” statement follows from Lemma 3.1.
(iii) By part (i) above, it suffices to show that there is a path from 𝑥 to 𝑦 in 𝑇(Γ(𝜇)) for any 𝑥𝑇(𝜇) and 𝑦Tof(𝜇). By part (ii) above, there exist adjacent vertices 𝑐 and 𝑑 in Tor(Γ(𝜇)) and Tof(Γ(𝜇)), respectively. Since Tor(Γ(𝜇)) is connected, there is a path from 𝑥 to 𝑐 in Tor(Γ(𝜇)), and, since Tof(Γ(𝜇)) is connected, there is a path from 𝑑 to 𝑦 in Tof(Γ(𝜇)). Then there is a path from 𝑥 to 𝑦 in 𝑇(Γ(𝜇)) since 𝑐 and 𝑑 are adjacent in 𝑇(Γ(𝜇)). Thus, 𝑇(Γ(𝜇)) is connected.

Proposition 3.3. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is not a submodule of 𝑀. If the identity of the ring 𝑅 is a sum of 𝑛 zero divisors, then every element of the 𝑀 is the sum of at most 𝑛𝜇-torsion elements.

Proof. Let 𝑥𝑀 and 𝑟𝑍(𝑅). We may assume that 𝜇(𝑥)𝜇(0𝑀). Then there is a nonzero element 𝑏𝑅 such that 𝑟𝑏=0, so 𝜇(𝑏(𝑟𝑥))=𝜇((𝑟𝑏)𝑥)=𝜇(0𝑀) with 𝜇(𝑟𝑥)𝜇(0𝑀). Therefore, if 𝑥𝑀 and 𝑟𝑅, then 𝑟𝑥𝑇(𝜇), so, for all 𝑥𝑀, 1=𝑐1++𝑐𝑛 implies that 𝑥=𝑐1𝑥++𝑐𝑛𝑥, as needed.

Theorem 3.4. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is not a submodule of 𝑀. Then 𝑇(Γ(𝜇)) is connected if and only if 𝑀 is generated by its 𝜇-torsion elements.

Proof. Let us first prove that the connectedness of the graph 𝑇(Γ(𝜇)) implies that the module 𝑀 is generated by its 𝜇-torsion elements. Suppose that this is not true. Then there exists 𝑥𝑀 which does not have a representation of the form 𝑥=𝑥1++𝑥𝑛, where 𝑥𝑖𝑇(𝜇). Moreover, 𝑥0 since 0𝑇(𝜇). We show that there does not exist a path from 0 to 𝑥 in 𝑇(Γ(𝜇)). If 0𝑦1𝑦2𝑦𝑚𝑥 is a path in 𝑇(Γ(𝜇)), 𝑦1,𝑦1+𝑦2,,𝑦𝑚1+𝑦𝑚,𝑦𝑚+𝑥 are 𝜇-torsion elements and 𝑥 may be represented as 𝑥=(𝑦𝑚+𝑥)(𝑦𝑚1+𝑦𝑚)++(1)𝑚1(𝑦1+𝑦2)+(1)𝑚𝑦1. This contradicts the assumption that 𝑥 is not a sum of 𝜇-torsion elements. The reverse implication may be proved in a similar way as in [6, Theorem  3.2].

We give here with an interesting result linking the 𝐿-torsion element graph 𝑇(Γ(𝜇)) to the total graph of a commutative ring 𝑇(Γ(𝑅)).

Theorem 3.5. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀). If 𝑇(Γ(𝑅)) is connected, then 𝑇(Γ(𝜇)) is a connected graph. In particular, 𝑑𝜇(0,𝑥)𝑑(0,1) for every 𝑥𝑀.

Proof. Note that, if 𝑥𝑀 and 𝑟𝑍(𝑅), then 𝑟𝑚𝑇(𝜇) (see Proposition 3.3). Now suppose that 𝑇(Γ(𝑅)) is connected, and let 𝑥𝑀. Let 0𝑠1𝑠2𝑠𝑛1 be a path from 0 to 1 in 𝑇(Γ(𝑅)). Then 𝑠1,𝑠1+𝑠2,,𝑠𝑛+1𝑍(𝑅); hence, 0𝑀𝑠1𝑥𝑠𝑛𝑥𝑥 is a path from 0𝑀 to 𝑥. As all vertices may be connected via 0𝑀, 𝑇(Γ(𝜇)) is connected.

Theorem 3.6. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is not a submodule of 𝑀. If every element of 𝑀 is a sum of at most 𝑛𝜇-torsion elements, then diam(𝑇(Γ(𝜇)))𝑛. If 𝑛 is the smallest such number, then diam(𝑇(Γ(𝜇)))=𝑛.

Proof. We first show that, by assumption, 𝑑𝜇(0,𝑥)𝑛 for every nonzero element 𝑥 of 𝑀. Assume that 𝑥=𝑥1++𝑥𝑛, where 𝑥𝑖𝑇(𝜇). Set 𝑦𝑖=(1)𝑛+𝑖(𝑥1++𝑥𝑛) for 𝑖=1,,𝑛. Then 0𝑦1𝑦2𝑦𝑛=𝑥 is a path from 0 to 𝑥 of length 𝑛 in 𝑇(Γ(𝜇)). Let 𝑢 and 𝑤 be distinct elements in 𝑀. We show that 𝑑𝜇(𝑢,𝑤)𝑛. If (𝑢𝑤)𝑧1𝑧2𝑧𝑛1 is a path from 0 to 𝑢𝑤 and 𝑢+𝑤𝑠1𝑠2𝑠𝑛1 is a path from 0 to 𝑢+𝑤, then, from the previous discussion, the lengths of both paths are at most 𝑛. Depending on the fact whether 𝑛 is even or odd, we obtain the paths 𝑧𝑢1𝑧𝑤2𝑧+𝑤𝑛1𝑤𝑤(3.1) or 𝑢(𝑠1+𝑤)(𝑠2𝑤)(𝑠𝑛1𝑤)𝑤 from 𝑢 to 𝑤 of length 𝑛. Assume that 𝑛 is the smallest such number, and let 𝑎=𝑎1+𝑎2++𝑎𝑛 be the shortest representation of the elements 𝑥 as a sum of 𝜇-torsion elements. From the previous discussion, we have 𝑑𝜇(0,𝑥)𝑛. Suppose that 𝑑𝜇(0,𝑥)=𝑘𝑛, and let 0𝑡1𝑡2𝑡𝑘1𝑥 be a path in 𝑇(Γ(𝜇)). It means, a presentation of the element 𝑥 as a sum of 𝑘<𝑛𝜇-torsion elements (see the proof of Theorem 3.4), which is a contradiction. This completes the proof.

Corollary 3.7. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑍(𝑅) is not an ideal of 𝑅 and <𝑍(𝑅)>=𝑅. If diam𝑇((Γ(𝑅)))=𝑛, then diam𝑇((Γ(𝜇)))𝑛. In particular, if 𝑅 is finite, then diam𝑇((Γ(𝜇)))2.

Proof. This follows from Proposition 3.3 and Theorem 3.6. Finally, if 𝑅 is a finite ring such that 𝑍(𝑅) is not an ideal of 𝑅, then diam𝑇((Γ(𝑅)))=2 by [3, Theorem  3.4], as required.

By Lemma 3.1, the following theorem may be proved in a similar way as in [6, Theorem  3.5].

Theorem 3.8. Let 𝑀 be a module over a ring 𝑅, and let 𝜇𝐿(𝑀) such that 𝑇(𝜇) is not a submodule of 𝑀. Then one has the following.(i)Either gr(Tor(Γ(𝜇)))=3 or gr(Tor(Γ(𝜇)))=.(ii)gr(𝑇(Γ(𝜇)))=3 if and only if gr(Tor(Γ(𝜇)))=3.(iii)If gr(𝑇(Γ(𝜇)))=4, then gr(Tor(Γ(𝜇)))=.(iv)If Char(𝑅)2, then gr(Tof(Γ(𝜇)))=3,4 or .

Example 3.9. Let 𝑀=𝑅=𝑍6 denote the ring of integers modulo 6. We define the mapping 𝜇𝑀[0,1] by 1𝜇(𝑥)=if𝑥=10,4otherwise.(3.2) Then 𝜇𝐿(𝑀) and 𝑇(𝜇)={0,2,3,4}. Now one can easily show that 𝑇(𝜇) is not a submodule of 𝑀 and Tof(𝜇)={1,5}. Clearly, Tor(Γ(𝜇)) is connected with diam(Tor(Γ(𝜇)))=2. Moreover, since 1+3𝑇(𝜇), we conclude that the subgraphs Tof(Γ(𝜇)) and Tor(Γ(𝜇)) of 𝑇(Γ(𝜇)) are not disjoint. Furthermore, 𝑇(Γ(𝜇)) is connected since Tof(Γ(𝜇)) is connected.