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ISRN Discrete Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 491936, 10 pages
http://dx.doi.org/10.5402/2011/491936
Research Article

The 𝐿-Total Graph of an 𝐿-Module

Department of Computer Engineering, University of Guilan, P.O. Box 3756, Rasht 41996-13769, Iran

Received 30 August 2011; Accepted 5 October 2011

Academic Editor: R.Β Yeh

Copyright Β© 2011 Reza Ebrahimi Atani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [2–5]). Let 𝑅 be a commutative ring with 𝑍(𝑅)being its set of zero-divisors elements. The total graph of 𝑅, denoted by 𝑇(Ξ“(𝑅)), is the (undirected) graph with all elements of 𝑅 as vertices, and, for distinct π‘₯,π‘¦βˆˆπ‘…, the vertices π‘₯ and 𝑦 are adjacent if and only if π‘₯+π‘¦βˆˆπ‘(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 Ξ“, by 𝐸(Ξ“) and 𝑉(Ξ“), 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 π‘Ž and 𝑏, denoted by 𝑑(π‘Ž,𝑏), is the length of the shortest path connecting them (if such a path does not exist, then 𝑑(π‘Ž,π‘Ž)=0 and 𝑑(π‘Ž,𝑏)=∞). The diameter of a graph Ξ“, denoted by diam(Ξ“), is equal to sup{𝑑(π‘Ž,𝑏)βˆΆπ‘Ž,π‘βˆˆπ‘‰(Ξ“)}. A graph is complete if it is connected with diameter less than or equal to one. The girth of a graph Ξ“, denoted gr(Ξ“), is the length of the shortest cycle in Ξ“, provided Ξ“ contains a cycle; otherwise, gr(Ξ“)=∞. We denote the complete graph on 𝑛 vertices by 𝐾𝑛 and the complete bipartite graph on π‘š and 𝑛 vertices by πΎπ‘š,𝑛 (we allow π‘š and 𝑛 to be infinite cardinals). We will sometimes call a 𝐾1,π‘š a star graph. We say that two (induced) subgraphs Ξ“1 and Ξ“2 of Ξ“ are disjoint if Ξ“1 and Ξ“2 have no common vertices and no vertex of Ξ“1 (resp., Ξ“2) is adjacent (in Ξ“) to any vertex not in Ξ“1 (resp., Ξ“2).

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 πœ‡ of a nonempty set 𝑋, we mean a function πœ‡ from 𝑋 to 𝐿. If 𝐿=[0,1], then πœ‡ 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 πœ‡βˆΆπ‘…β†’πΏ, where (𝑅,+,.) is a ring, which satisfies the following.(1)πœ‡β‰ 0.(2)πœ‡(π‘₯βˆ’π‘¦)β‰₯πœ‡(π‘₯)βˆ§πœ‡(𝑦) for every π‘₯,𝑦 in 𝑅.(3)πœ‡(π‘₯𝑦)β‰₯πœ‡(π‘₯)βˆ¨πœ‡(𝑦) for every π‘₯,𝑦 in 𝑅.

Definition 1.2. Let πœ‡βˆˆπΏπ‘…. Then πœ‡ is called an 𝐿-ideal of 𝑅 if for every π‘₯,π‘¦βˆˆπ‘… the following conditions are satisfied. (1)πœ‡(π‘₯βˆ’π‘¦)β‰₯πœ‡(π‘₯)βˆ§πœ‡(𝑦).(2)πœ‡(π‘₯𝑦)β‰₯πœ‡(π‘₯)βˆ¨πœ‡(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 πœ‡βˆˆπΏπ‘€. Then πœ‡ is called an 𝐿-fuzzy 𝑅-module of 𝑀 if for all π‘₯,π‘¦βˆˆπ‘€ and for all π‘Ÿβˆˆπ‘… the following conditions are satisfied.(1)πœ‡(π‘₯βˆ’π‘¦)β‰₯πœ‡(π‘₯)βˆ§πœ‡(𝑦).(2)πœ‡(π‘Ÿπ‘₯)β‰₯πœ‡(π‘₯).(3)πœ‡(0𝑀)=πœ‡(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 πœ‡βˆˆπΏ(𝑀). Then πœ‡(π‘š)β‰€πœ‡(0𝑀) for every π‘šβˆˆπ‘€.

2. 𝑇(πœ‡) Is a Submodule of 𝑀

Let 𝑀 be a module over a commutative ring 𝑅, and let πœ‡βˆˆπΏ(𝑀). The structure of the 𝐿-total torsion element graph 𝑇(Ξ“(πœ‡)) may be completely described in those cases when πœ‡-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 πœ‡βˆˆπΏ(𝑀). A πœ‡-torsion element is an element π‘šβˆˆπ‘€ with πœ‡(π‘š)β‰ πœ‡(0𝑀) for which there exists a nonzero element π‘Ÿ of 𝑅 such that πœ‡(π‘Ÿπ‘š)=πœ‡(0𝑀).

The set of πœ‡-torsion elements in 𝑀 will be denoted by 𝑇(πœ‡).

Definition 2.2. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀). We define the 𝐿-total torsion element graph of an 𝐿-module 𝑇(Ξ“(πœ‡)) as follows: 𝑉(𝑇(Ξ“(πœ‡)))=𝑀, 𝐸(𝑇(Ξ“(πœ‡)))={{π‘₯,𝑦}∢π‘₯+π‘¦βˆˆπ‘‡(πœ‡)}.

Notation 1. For the πœ‡-torsion element graph 𝑇(Ξ“(πœ‡)), we denote the diameter, the girth, and the distance between two distinct vertices π‘Ž and 𝑏, by diam(𝑇(Ξ“(πœ‡))), gr(𝑇(Ξ“(πœ‡))), and π‘‘πœ‡(π‘Ž,𝑏), respectively.

Remark 2.3. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀). Clearly, if πœ‡ is a nonzero constant, then 𝑇(Ξ“(πœ‡))=βˆ…. So throughout this paper, we will assume, unless otherwise stated, that πœ‡ is not a nonzero constant. Thus, there is a nonzero element 𝑦 of 𝑀 such that πœ‡(𝑦)β‰ πœ‡(0𝑀).

We will use Tof(πœ‡) to denote the set of elements of 𝑀 that are not πœ‡-torsion elements. Let Tof(Ξ“(πœ‡)) be the (induced) subgraph of 𝑇(Ξ“(πœ‡)) with vertices Tof(πœ‡), and let Tor(Ξ“(πœ‡)) be the (induced) subgraph of 𝑇(Ξ“(πœ‡)) with vertices 𝑇(πœ‡).

Definition 2.4. Let 𝑀 be a module over a ring 𝑅, and πœ‡βˆˆπΏ(𝑀). One defines the set annπœ‡(𝑀) by annπœ‡(𝑀)={π‘Ÿβˆˆπ‘…βˆΆπœ‡(π‘Ÿπ‘€)={πœ‡(0𝑀)}}, the πœ‡-annihilator of 𝑀.

Lemma 2.5. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀). Then annπœ‡(𝑀) is an 𝐿-ideal of 𝑅.

Proof. Let π‘Ÿ,π‘ βˆˆannπœ‡(𝑀) and π‘‘βˆˆπ‘…. If π‘šβˆˆπ‘€, then we have πœ‡((π‘Ÿβˆ’π‘ )π‘š)β‰₯πœ‡(π‘Ÿπ‘š)βˆ§πœ‡(βˆ’π‘ π‘š)=πœ‡(0𝑀)βˆ§πœ‡(0𝑀)=πœ‡(0𝑀) and πœ‡(π‘‘π‘Ÿπ‘š)=πœ‡(𝑑(π‘Ÿπ‘š))β‰₯πœ‡(π‘Ÿπ‘š)=πœ‡(0𝑀). It then follows from Lemma 1.4 that πœ‡((π‘Ÿβˆ’π‘ )π‘š)=πœ‡(0𝑀); hence π‘Ÿβˆ’π‘ βˆˆannπœ‡(𝑀). Similarly, π‘Ÿπ‘‘βˆˆannπœ‡(𝑀).

Theorem 2.6. Let 𝑀 be a module over a ring 𝑅 and let πœ‡βˆˆπΏ(𝑀). Then the 𝐿-torsion element graph 𝑇(Ξ“(πœ‡)) is complete if and only if 𝑇(πœ‡)=𝑀.

Proof. If 𝑇(πœ‡)=𝑀, then for any vertices π‘š,π‘šξ…žβˆˆπ‘€, one has π‘š+π‘šξ…žβˆˆπ‘‡(πœ‡); hence they are adjacent in 𝑇(Ξ“(πœ‡)). On the other hand, if 𝑇(Ξ“(πœ‡)) is complete, then every vertex is adjacent to 0. Thus,π‘š=π‘š+0βˆˆπ‘‡(πœ‡) for every π‘šβˆˆπ‘€. This completes the proof.

Theorem 2.7. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a submodule of 𝑀. Then one has the following.(i)Tor(Ξ“(πœ‡)) is a complete (induced) subgraph of 𝑇(Ξ“(πœ‡)) and Tor(Ξ“(πœ‡)) is disjoint from Tof(Ξ“(πœ‡)).(ii)If annΞΌ(𝑀)β‰ 0, then 𝑇(Ξ“(πœ‡)) is a complete graph.

Proof. (i) Tor(Ξ“(πœ‡)) is complete directly from the definition. Finally, if π‘šβˆˆπ‘‡(πœ‡) and π‘šξ…žβˆˆTof(πœ‡) were adjacent, then π‘š+π‘šξ…žβˆˆπ‘‡(πœ‡); so this, since 𝑇(πœ‡) is a submodule, would lead to the contradiction π‘šξ…žβˆˆπ‘‡(πœ‡).
(ii) Let π‘šβˆˆπ‘€. we may assume that πœ‡(π‘š)β‰ πœ‡(0𝑀). By assumption, there exists 0β‰ π‘ βˆˆπ‘… with πœ‡(𝑠𝑀)=πœ‡(0𝑀), so πœ‡(π‘ π‘š)=πœ‡(0𝑀). Thus π‘šβˆˆπ‘‡(πœ‡), and; therefore, 𝑇(Ξ“(πœ‡)) is a complete graph by Theorem 2.6.

Theorem 2.8. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀). Then 𝑇(Ξ“(πœ‡)) is totally disconnected if and only if 𝑅 has characteristic 2 and 𝑇(πœ‡)={0𝑀}.

Proof. If 𝑇(πœ‡)={0𝑀}, then the vertices π‘š1 and π‘š2 are adjacent if and only if π‘š1=βˆ’π‘š2. Then 𝑇(Ξ“(πœ‡)) is a disconnected graph, and its only edges are those that connect vertices π‘šπ‘– and βˆ’π‘šπ‘– (we do not need a priori assumption that 𝑅 has characteristic 2). Conversely, assume that 𝑇(Ξ“(πœ‡)) is totally disconnected. Then 0+π‘šβˆ‰π‘‡(πœ‡) for every nonzero element π‘š of 𝑀. Thus, 𝑇(πœ‡)={0𝑀}. Further, since π‘š+(βˆ’π‘š)=0, we have π‘š=βˆ’π‘š (so πœ‡(2π‘š)=πœ‡(0𝑀)) for every π‘šβˆˆπ‘€ with πœ‡(π‘š)β‰ πœ‡(0𝑀) by the total disconnectedness of the graph 𝑇(Ξ“(πœ‡)). As 𝑇(πœ‡)={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 πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a submodule of 𝑀. If π‘šβˆˆTof(πœ‡), then 2π‘šβˆˆπ‘‡(πœ‡) if and only if 2βˆˆπ‘(𝑅).

Proof. First suppose that 2π‘šβˆˆπ‘‡(πœ‡). Since π‘šβˆ‰π‘‡(πœ‡), we get that πœ‡(π‘š)β‰ πœ‡(0𝑀), and, for all π‘Ÿβˆˆπ‘…,πœ‡(π‘Ÿπ‘š)=πœ‡(0𝑀) implies that π‘Ÿ=0. Since 2π‘šβˆˆπ‘‡(πœ‡), there is a nonzero element π‘βˆˆπ‘… such that πœ‡(𝑐(2π‘š))=πœ‡((2𝑐)π‘š)=πœ‡(0𝑀), and, since π‘šβˆ‰π‘‡(πœ‡), one must have 2𝑐=0; hence, 2βˆˆπ‘(𝑅). Conversely, assume that 2βˆˆπ‘(𝑅). Then there exists 0β‰ π‘‘βˆˆπ‘… with 2𝑑=0. Since πœ‡(0𝑀)=πœ‡((2𝑑)π‘š)=πœ‡(𝑑(2π‘š)), we have 2π‘šβˆˆπ‘‡(πœ‡).

Theorem 2.10. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a proper submodule of 𝑀. Then 𝑇(Ξ“(πœ‡)) is disconnected.

Proof. If 𝑇(πœ‡)={0𝑀}, then 𝑇(Ξ“(πœ‡)) is disconnected by Theorem 2.8. If 𝑇(πœ‡)β‰ {0𝑀}, then the subgraphs of Tor(Ξ“(πœ‡)) and Tof(Ξ“(πœ‡)) are disjoint by Theorem 2.7 (i), as required.

Theorem 2.11. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a proper submodule of M. Suppose |𝑇(πœ‡)|=𝛼 and |𝑀/𝑇(πœ‡)|=𝛽. Then one has the following.(i)If 2βˆˆπ‘(𝑅), then 𝑇(Ξ“(πœ‡)) is a union of 𝛽 disjoint complete graphs 𝐾𝛼.(ii)If 2βˆ‰π‘(𝑅), then 𝑇(Ξ“(πœ‡)) is a union of (π›½βˆ’1)/2 disjoint bipartite graphs 𝐾𝛼,𝛼 and one complete graph 𝐾𝛼.

Proof. (i) Assume that 2βˆˆπ‘(𝑅) and let π‘š,π‘šξ…žβˆˆTof(πœ‡) be such that π‘š+𝑇(πœ‡)β‰ π‘šξ…ž+𝑇(πœ‡). The elements π‘š+𝑑, π‘š+π‘‘ξ…ž from the same coset π‘š+𝑇(πœ‡) are adjacent if and only if 2π‘šβˆˆπ‘‡(πœ‡), so 2βˆˆπ‘(𝑅), according to the Proposition 2.9. Then π‘š+𝑑 and π‘šξ…ž+π‘‘ξ…ž are not adjacent (otherwise, we would have π‘šβˆ’π‘šξ…ž=π‘š+π‘šξ…žβˆ’2π‘šξ…žβˆˆπ‘‡(πœ‡)), and; therefore, π‘š+𝑇(πœ‡)=π‘šξ…ž+𝑇(πœ‡). Since every coset has cardinality 𝛼, we conclude that 𝑇(Ξ“(πœ‡)) is the disjoint union of 𝛽 complete graph 𝐾𝛼.
(ii) If 2βˆ‰π‘(𝑅), then the elements π‘š+𝑑, π‘š+π‘‘ξ…ž from π‘š+𝑇(πœ‡) are obviously not adjacent. The elements π‘š+𝑑, π‘šξ…ž+π‘‘ξ…ž from different cosets are adjacent if and only if π‘š+π‘šξ…žβˆˆπ‘‡(πœ‡) or π‘š+𝑇(πœ‡)=(βˆ’π‘š)+𝑇(πœ‡). In this way we obtain that the subgraph spanned by the vertices from Tof(πœ‡) is a disjoint union of (π›½βˆ’1)/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 πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a proper submodule of 𝑀. Then one has the following.(i)Tof(Ξ“(πœ‡)) is complete if and only if either |𝑀/𝑇(πœ‡)|=2 or |𝑀/𝑇(πœ‡)|=|𝑀|=3.(ii)Tof(Ξ“(πœ‡)) is connected if and only if either |𝑀/𝑇(πœ‡)|=2 or |𝑀/𝑇(πœ‡)|=3.(iii)Tof(Ξ“(πœ‡)) and, hence; (Tor(Ξ“(πœ‡)) and 𝑇(Ξ“(πœ‡))) is totally disconnected if and only if 𝑇(πœ‡)={0𝑀} and 2βˆˆπ‘(𝑅).

Proof. Let |𝑀/𝑇(πœ‡)|=𝛽 and |𝑇(πœ‡)|=𝛼.(i)Let Tof(Ξ“(πœ‡)) be complete. Then, by Theorem 2.11, Tof(Ξ“(πœ‡)) is complete if and only if Tof(Ξ“(πœ‡)) is a single 𝐾𝛼 or 𝐾1,1. If 2βˆˆπ‘(𝑅), then π›½βˆ’1=1. Thus, 𝛽=2, and hence |𝑀/𝑇(πœ‡)|=2. If 2βˆ‰π‘(𝑅), then 𝛼=1 and (π›½βˆ’1)/2=1. Thus, 𝑇(πœ‡)={0} and 𝛽=3; hence, |𝑀|=|𝑀/𝑇(πœ‡)|=3. The reverse implication may be proved in a similar way as in [6, Theorem  2.6 (1)].(ii)By theorem 2.11, Tof(Ξ“(πœ‡)) is connected if and only if Tof(Ξ“(πœ‡)) is a single 𝐾𝛼 or 𝐾𝛼,𝛼. Thus, either π›½βˆ’1=1 if 2βˆˆπ‘(𝑅) or (π›½βˆ’1)/2=1 if 2βˆ‰π‘(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(Ξ“(πœ‡)) is totally disconnected if and only if it is a disjoint union of 𝐾1’s. So by Theorem 2.11, |𝑇(πœ‡)|=1 and |𝑀/𝑇(πœ‡)|=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(Ξ“(πœ‡)).

Theorem 2.13. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a proper submodule of 𝑀. Then one has the following.(i)diam(Tof(Ξ“(πœ‡)))=0 if and only if 𝑇(πœ‡)={0} and |𝑀|=2.(ii)diam(Tof(Ξ“(πœ‡)))=1 if and only if either 𝑇(πœ‡)β‰ {0𝑀} and |𝑀/𝑇(πœ‡)|=2 or 𝑇(πœ‡)={0} and |𝑀|=3.(iii)diam(Tof(Ξ“(πœ‡)))=2 if and only if 𝑇(πœ‡)β‰ {0𝑀} and |𝑀/𝑇(πœ‡)|=3.(iv)Otherwise, diam(Tof(Ξ“(πœ‡)))=∞.

Proposition 2.14. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a proper submodule of 𝑀. Then gr(Tof(Ξ“(πœ‡)))=3,4 or ∞. In particular, gr(Tof(Ξ“(πœ‡)))≀4 if Tof(Ξ“(πœ‡)) contains a cycle.

Proof. Let Tof(Ξ“(πœ‡)) contain a cycle. Then since Tof(Ξ“(πœ‡)) 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(Ξ“(πœ‡)))≀4.

Theorem 2.15. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is a proper submodule of 𝑀. Then one has the following.(i)(a)gr(Tof(Ξ“(πœ‡)))=3 if and only if 2βˆˆπ‘(𝑅) and |𝑇(πœ‡)|β‰₯3.(b)gr(Tof(Ξ“(πœ‡)))=4 if and only if 2βˆ‰π‘(𝑅) and |𝑇(πœ‡)|β‰₯2.(c)Otherwise, gr(Tof(Ξ“(πœ‡)))=∞.(ii)(a)gr(𝑇(Ξ“(πœ‡)))=3 if and only if |𝑇(πœ‡)|β‰₯3.(b)gr(𝑇(Ξ“(πœ‡)))=4 if and only if 2βˆ‰π‘(𝑅) and |𝑇(πœ‡)|=2.(c)Otherwise, gr(𝑇(Ξ“(πœ‡)))=∞.

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(πœ‡) is a submodule. The question under what conditions 𝑇(πœ‡) 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 πœ‡βˆˆπΏ(𝑀). Then one has the following.(i)If 𝑍(𝑅)={0𝑅}, then 𝑇(πœ‡) 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(πœ‡) is a submodule of 𝑀.

Proof. (i) Let π‘š,π‘šξ…žβˆˆπ‘‡(πœ‡) and π‘Ÿβˆˆπ‘…. There are nonzero elements π‘Ž,π‘βˆˆπ‘… such that πœ‡(π‘š)β‰ πœ‡(0𝑀), πœ‡(π‘šξ…ž)β‰ πœ‡(0𝑀), and πœ‡(π‘Žπ‘š)=πœ‡(π‘π‘šξ…ž)=πœ‡(0𝑀) with π‘Žπ‘β‰ 0 (since 𝑅 is an integral domain). It follows that πœ‡(π‘Žπ‘(π‘š+π‘šξ…ž))β‰₯πœ‡(π‘Žπ‘π‘š)βˆ§πœ‡(π‘Žπ‘π‘šξ…ž)=πœ‡(0𝑀)βˆ§πœ‡(0𝑀)=πœ‡(0𝑀); hence, πœ‡(π‘Žπ‘(π‘š+π‘šξ…ž))=πœ‡(0𝑀) by Lemma 1.4. Thus, π‘š+π‘šξ…žβˆˆπ‘‡(πœ‡). Similarly, π‘Ÿπ‘šβˆˆπ‘‡(πœ‡), and this completes the proof.
(ii) Assume that 𝑇(πœ‡) is not a submodule of 𝑀. Then there are elements π‘š,π‘šξ…žβˆˆπ‘‡(πœ‡) such that π‘š+π‘šξ…žβˆ‰π‘‡(πœ‡). By assumption, there exist nonzero elements π‘Ÿ,π‘ βˆˆπ‘… such that πœ‡(π‘Ÿπ‘š)=πœ‡(0𝑀)=πœ‡(π‘ π‘šβ€²)=πœ‡(0𝑀), where πœ‡(π‘š)β‰ πœ‡(0𝑀) and πœ‡(π‘šβ€²)β‰ πœ‡(0𝑀). Then πœ‡(π‘Ÿπ‘ (π‘š+π‘šξ…ž))=πœ‡(0𝑀) and π‘š+π‘šξ…žβˆ‰π‘‡(πœ‡), so we must have π‘Ÿπ‘ =0, and; thus, π‘Ÿ,π‘ βˆˆπ‘(𝑅). Since 𝑐 is nilpotent, we have π‘Ÿ=π‘Ÿ1𝑐𝑑 and 𝑠=𝑠1𝑐𝑒, for some π‘Ÿ1,𝑠1βˆ‰π‘(𝑅). We may assume that 𝑑β‰₯𝑒. Then for the nonzero element 𝑠1π‘Ÿ of 𝑅 we have πœ‡(𝑠1π‘Ÿ(π‘š+π‘šξ…ž))=πœ‡(0𝑀) which is contrary to the assumption that π‘š+π‘šξ…žβˆ‰π‘‡(πœ‡).

Example 2.17. Assume that 𝑅=β„€ is the ring integers, and let 𝑀=𝑅. We define the mapping πœ‡βˆΆπ‘€β†’[0,1] by ⎧βŽͺ⎨βŽͺ⎩1πœ‡(π‘š)=2if1π‘₯∈2β„€,5otherwise.(2.1) Then πœ‡βˆˆπΏ(𝑀) and 𝑇(πœ‡)=𝑀. Thus, 𝑇(Ξ“(πœ‡)) 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 πœ‡1βˆΆπ‘€1β†’[0,1] by πœ‡1ξƒ―1(π‘₯)=ifπ‘₯=10,2otherwise(2.2) and πœ‡2βˆΆπ‘€2β†’[0,1] by πœ‡2(ξƒ―1π‘š)=ifπ‘₯=10,3otherwise.(2.3) Then, for each 𝑖 (1≀𝑖≀2), πœ‡π‘–βˆˆπΏ(𝑀𝑖), 𝑇(πœ‡1)={0,2,4,6}, and 𝑇(πœ‡2)={0,5,10,15,20}. An inspection will show that 𝑇(πœ‡1) and 𝑇(πœ‡2) are submodules of 𝑀1 and 𝑀2, respectively. Therefore, by Theorem 2.11, we have the following results.(1)Since 2βˆˆπ‘(𝑅1), we conclude that 𝑇(Ξ“(πœ‡1)) is a union of 2 disjoint 𝐾4.(2)Since 2βˆ‰π‘(𝑅2), we conclude that 𝑇(Ξ“(πœ‡2)) is a disjoint union of 2 complete graph 𝐾5 and 5 bipartite 𝐾5,5.

3. 𝑇(πœ‡) 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 πœ‡βˆˆπΏ(𝑀). In this section, we study the 𝐿-torsion element graph 𝑇(Ξ“(πœ‡)) when 𝑇(πœ‡) is not a submodule of 𝑀.

Lemma 3.1. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is not a submodule of 𝑀. Then there are distinct π‘š,π‘šξ…žβˆˆπ‘‡(πœ‡)βˆ— such that π‘š+π‘šξ…žβˆˆTof(πœ‡).

Proof. It suffices to show that 𝑇(πœ‡) is always closed under scalar multiplication of its elements by elements of 𝑅. Let π‘šβˆˆπ‘‡(πœ‡) and π‘Ÿβˆˆπ‘…. There is a nonzero element π‘ βˆˆπ‘… with πœ‡(π‘ π‘š)=πœ‡(0𝑀) such that πœ‡(π‘š)β‰ πœ‡(0𝑀), so πœ‡(𝑠(π‘Ÿπ‘š))=πœ‡(π‘Ÿ(π‘ π‘š))β‰₯πœ‡(π‘ π‘š)=πœ‡(0𝑀); hence, πœ‡(𝑠(π‘Ÿπ‘š))=πœ‡(0𝑀) by Lemma 1.4, as required.

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

Proof. (i) Let π‘₯βˆˆπ‘‡(πœ‡)βˆ—. Then π‘₯ is adjacent to 0. Thus, π‘₯βˆ’0βˆ’π‘¦ is a path in Tor(Ξ“(πœ‡)) of length two between any two distinct π‘₯,π‘¦βˆˆπ‘‡(πœ‡)βˆ—. Moreover, there exist nonadjacent π‘₯,π‘¦βˆˆπ‘‡(πœ‡)βˆ— by Lemma 3.1; thus, diam(Tor(Ξ“(πœ‡)))=2.
(ii) By Lemma 3.1, there exist distinct π‘₯,π‘¦βˆˆπ‘‡(πœ‡)βˆ— such that π‘₯+π‘¦βˆˆTof(πœ‡). Then βˆ’π‘₯βˆˆπ‘‡(πœ‡) and π‘₯+π‘¦βˆˆTof(πœ‡) are adjacent vertices in 𝑇(Ξ“(πœ‡)) since βˆ’π‘₯+(π‘₯+𝑦)=π‘¦βˆˆπ‘‡(πœ‡). Finally, the β€œin particular” statement follows from Lemma 3.1.
(iii) By part (i) above, it suffices to show that there is a path from π‘₯ to 𝑦 in 𝑇(Ξ“(πœ‡)) for any π‘₯βˆˆπ‘‡(πœ‡) and π‘¦βˆˆTof(πœ‡). By part (ii) above, there exist adjacent vertices 𝑐 and 𝑑 in Tor(Ξ“(πœ‡)) and Tof(Ξ“(πœ‡)), respectively. Since Tor(Ξ“(πœ‡)) is connected, there is a path from π‘₯ to 𝑐 in Tor(Ξ“(πœ‡)), and, since Tof(Ξ“(πœ‡)) is connected, there is a path from 𝑑 to 𝑦 in Tof(Ξ“(πœ‡)). Then there is a path from π‘₯ to 𝑦 in 𝑇(Ξ“(πœ‡)) since 𝑐 and 𝑑 are adjacent in 𝑇(Ξ“(πœ‡)). Thus, 𝑇(Ξ“(πœ‡)) is connected.

Proposition 3.3. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) 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 π‘›πœ‡-torsion elements.

Proof. Let π‘₯βˆˆπ‘€ and π‘Ÿβˆˆπ‘(𝑅). We may assume that πœ‡(π‘₯)β‰ πœ‡(0𝑀). Then there is a nonzero element π‘βˆˆπ‘… such that π‘Ÿπ‘=0, so πœ‡(𝑏(π‘Ÿπ‘₯))=πœ‡((π‘Ÿπ‘)π‘₯)=πœ‡(0𝑀) with πœ‡(π‘Ÿπ‘₯)β‰ πœ‡(0𝑀). Therefore, if π‘₯βˆˆπ‘€ and π‘Ÿβˆˆπ‘…, then π‘Ÿπ‘₯βˆˆπ‘‡(πœ‡), so, for all π‘₯βˆˆπ‘€, 1=𝑐1+β‹―+𝑐𝑛 implies that π‘₯=𝑐1π‘₯+β‹―+𝑐𝑛π‘₯, as needed.

Theorem 3.4. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is not a submodule of 𝑀. Then 𝑇(Ξ“(πœ‡)) is connected if and only if 𝑀 is generated by its πœ‡-torsion elements.

Proof. Let us first prove that the connectedness of the graph 𝑇(Ξ“(πœ‡)) implies that the module 𝑀 is generated by its πœ‡-torsion elements. Suppose that this is not true. Then there exists π‘₯βˆˆπ‘€ which does not have a representation of the form π‘₯=π‘₯1+β‹―+π‘₯𝑛, where π‘₯π‘–βˆˆπ‘‡(πœ‡). Moreover, π‘₯β‰ 0 since 0βˆˆπ‘‡(πœ‡). We show that there does not exist a path from 0 to π‘₯ in 𝑇(Ξ“(πœ‡)). If 0βˆ’π‘¦1βˆ’π‘¦2βˆ’β‹―βˆ’π‘¦π‘šβˆ’π‘₯ is a path in 𝑇(Ξ“(πœ‡)), 𝑦1,𝑦1+𝑦2,…,π‘¦π‘šβˆ’1+π‘¦π‘š,π‘¦π‘š+π‘₯ are πœ‡-torsion elements and π‘₯ may be represented as π‘₯=(π‘¦π‘š+π‘₯)βˆ’(π‘¦π‘šβˆ’1+π‘¦π‘š)+β‹―+(βˆ’1)π‘šβˆ’1(𝑦1+𝑦2)+(βˆ’1)π‘šπ‘¦1. This contradicts the assumption that π‘₯ is not a sum of πœ‡-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(Ξ“(πœ‡)) to the total graph of a commutative ring 𝑇(Ξ“(𝑅)).

Theorem 3.5. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀). If 𝑇(Ξ“(𝑅)) is connected, then 𝑇(Ξ“(πœ‡)) is a connected graph. In particular, π‘‘πœ‡(0,π‘₯)≀𝑑(0,1) for every π‘₯βˆˆπ‘€.

Proof. Note that, if π‘₯βˆˆπ‘€ and π‘Ÿβˆˆπ‘(𝑅), then π‘Ÿπ‘šβˆˆπ‘‡(πœ‡) (see Proposition 3.3). Now suppose that 𝑇(Ξ“(𝑅)) is connected, and let π‘₯βˆˆπ‘€. Let 0βˆ’π‘ 1βˆ’π‘ 2βˆ’β‹―βˆ’π‘ π‘›βˆ’1 be a path from 0 to 1 in 𝑇(Ξ“(𝑅)). Then 𝑠1,𝑠1+𝑠2,…,𝑠𝑛+1βˆˆπ‘(𝑅); hence, 0π‘€βˆ’π‘ 1π‘₯βˆ’β‹―βˆ’π‘ π‘›π‘₯βˆ’π‘₯ is a path from 0𝑀 to π‘₯. As all vertices may be connected via 0𝑀, 𝑇(Ξ“(πœ‡)) is connected.

Theorem 3.6. Let 𝑀 be a module over a ring 𝑅, and let πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is not a submodule of 𝑀. If every element of 𝑀 is a sum of at most π‘›πœ‡-torsion elements, then diam(𝑇(Ξ“(πœ‡)))≀𝑛. If 𝑛 is the smallest such number, then diam(𝑇(Ξ“(πœ‡)))=𝑛.

Proof. We first show that, by assumption, π‘‘πœ‡(0,π‘₯)≀𝑛 for every nonzero element π‘₯ of 𝑀. Assume that π‘₯=π‘₯1+β‹―+π‘₯𝑛, where π‘₯π‘–βˆˆπ‘‡(πœ‡). Set 𝑦𝑖=(βˆ’1)𝑛+𝑖(π‘₯1+β‹―+π‘₯𝑛) for 𝑖=1,…,𝑛. Then 0βˆ’π‘¦1βˆ’π‘¦2βˆ’β‹―βˆ’π‘¦π‘›=π‘₯ is a path from 0 to π‘₯ of length 𝑛 in 𝑇(Ξ“(πœ‡)). Let 𝑒 and 𝑀 be distinct elements in 𝑀. We show that π‘‘πœ‡(𝑒,𝑀)≀𝑛. If (π‘’βˆ’π‘€)βˆ’π‘§1βˆ’π‘§2βˆ’β‹―βˆ’π‘§π‘›βˆ’1 is a path from 0 to π‘’βˆ’π‘€ and 𝑒+π‘€βˆ’π‘ 1βˆ’π‘ 2βˆ’β‹―βˆ’π‘ π‘›βˆ’1 is a path from 0 to 𝑒+𝑀, 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 ξ€·π‘§π‘’βˆ’1ξ€Έβˆ’ξ€·π‘§βˆ’π‘€2𝑧+π‘€βˆ’β‹―βˆ’π‘›βˆ’1ξ€Έβˆ’π‘€βˆ’π‘€(3.1) or π‘’βˆ’(𝑠1+𝑀)βˆ’(𝑠2βˆ’π‘€)βˆ’β‹―βˆ’(π‘ π‘›βˆ’1βˆ’π‘€)βˆ’π‘€ from 𝑒 to 𝑀 of length 𝑛. Assume that 𝑛 is the smallest such number, and let π‘Ž=π‘Ž1+π‘Ž2+β‹―+π‘Žπ‘› be the shortest representation of the elements π‘₯ as a sum of πœ‡-torsion elements. From the previous discussion, we have π‘‘πœ‡(0,π‘₯)≀𝑛. Suppose that π‘‘πœ‡(0,π‘₯)=π‘˜β‰€π‘›, and let 0βˆ’π‘‘1βˆ’π‘‘2βˆ’β‹―βˆ’π‘‘π‘˜βˆ’1βˆ’π‘₯ be a path in 𝑇(Ξ“(πœ‡)). It means, a presentation of the element π‘₯ as a sum of π‘˜<π‘›πœ‡-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 πœ‡βˆˆπΏ(𝑀) such that 𝑍(𝑅) is not an ideal of 𝑅 and <𝑍(𝑅)>=𝑅. If diam𝑇((Ξ“(𝑅)))=𝑛, then diam𝑇((Ξ“(πœ‡)))≀𝑛. In particular, if 𝑅 is finite, then diam𝑇((Ξ“(πœ‡)))≀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((Ξ“(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 πœ‡βˆˆπΏ(𝑀) such that 𝑇(πœ‡) is not a submodule of 𝑀. Then one has the following.(i)Either gr(Tor(Ξ“(πœ‡)))=3 or gr(Tor(Ξ“(πœ‡)))=∞.(ii)gr(𝑇(Ξ“(πœ‡)))=3 if and only if gr(Tor(Ξ“(πœ‡)))=3.(iii)If gr(𝑇(Ξ“(πœ‡)))=4, then gr(Tor(Ξ“(πœ‡)))=∞.(iv)If Char(𝑅)β‰ 2, then gr(Tof(Ξ“(πœ‡)))=3,4 or ∞.

Example 3.9. Let 𝑀=𝑅=𝑍6 denote the ring of integers modulo 6. We define the mapping πœ‡βˆΆπ‘€β†’[0,1] by ξƒ―1πœ‡(π‘₯)=ifπ‘₯=10,4otherwise.(3.2) Then πœ‡βˆˆπΏ(𝑀) and 𝑇(πœ‡)={0,2,3,4}. Now one can easily show that 𝑇(πœ‡) is not a submodule of 𝑀 and Tof(πœ‡)={1,5}. Clearly, Tor(Ξ“(πœ‡)) is connected with diam(Tor(Ξ“(πœ‡)))=2. Moreover, since 1+3βˆˆπ‘‡(πœ‡), we conclude that the subgraphs Tof(Ξ“(πœ‡)) and Tor(Ξ“(πœ‡)) of 𝑇(Ξ“(πœ‡)) are not disjoint. Furthermore, 𝑇(Ξ“(πœ‡)) is connected since Tof(Ξ“(πœ‡)) is connected.

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