The total torsion element graph of semimodules over commutative semirings
We introduce and investigate the total torsion element graph of semimodules over a commutative semiring with non-zero identity. The main purpose of this paper is to extend the definition and results given in [2] to more general semimodule case.
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nasplib_isofts_kiev_ua-123456789-1523032025-02-09T09:49:25Z The total torsion element graph of semimodules over commutative semirings Ebrahimi Atani, S. Esmaeili Khalil Saraei, F. We introduce and investigate the total torsion element graph of semimodules over a commutative semiring with non-zero identity. The main purpose of this paper is to extend the definition and results given in [2] to more general semimodule case. 2013 Article The total torsion element graph of semimodules over commutative semirings / S. Ebrahimi Atani, F. Esmaeili Khalil Saraei // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 1–15. — Бібліогр.: 15 назв. — англ. 1726-3255 2010 MSC:16Y60, 05C75. https://nasplib.isofts.kiev.ua/handle/123456789/152303 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We introduce and investigate the total torsion element graph of semimodules over a commutative semiring with non-zero identity. The main purpose of this paper is to extend the definition and results given in [2] to more general semimodule case. |
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Ebrahimi Atani, S. Esmaeili Khalil Saraei, F. |
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Ebrahimi Atani, S. Esmaeili Khalil Saraei, F. The total torsion element graph of semimodules over commutative semirings Algebra and Discrete Mathematics |
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Ebrahimi Atani, S. Esmaeili Khalil Saraei, F. |
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Ebrahimi Atani, S. |
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The total torsion element graph of semimodules over commutative semirings |
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The total torsion element graph of semimodules over commutative semirings |
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The total torsion element graph of semimodules over commutative semirings |
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The total torsion element graph of semimodules over commutative semirings |
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The total torsion element graph of semimodules over commutative semirings |
| title_sort |
total torsion element graph of semimodules over commutative semirings |
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Інститут прикладної математики і механіки НАН України |
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2013 |
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The total torsion element graph of semimodules over commutative semirings / S. Ebrahimi Atani, F. Esmaeili Khalil Saraei // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 1–15. — Бібліогр.: 15 назв. — англ. |
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Algebra and Discrete Mathematics |
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| first_indexed |
2025-11-25T12:53:57Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 1 – 15
© Journal “Algebra and Discrete Mathematics”
The total torsion element graph of semimodules
over commutative semirings
S. Ebrahimi Atani and F. Esmaeili Khalil Saraei
Communicated by D. Simson
Abstract. We introduce and investigate the total torsion
element graph of semimodules over a commutative semiring with
non-zero identity. The main purpose of this paper is to extend the
definition and results given in [2] to more general semimodule case.
1. Introduction
In [6], Beck associated to any commutative ring R its zero-divisor
graph G(R) whose vertices are the zero-divisors of R (including 0), with
two vertices a, b joined by an edge in case ab = 0. The problem Beck
studied was how to color the vertices of G(R) with the smallest number of
colors such that no two adjacent vertices in the graph had the same color.
Beck conjectured this number is the clique number of G(R), but Beck’s
question was settled in the negative in [1]. In [3], Anderson and Livingston
introduced and studied the subgraph Γ(R) whose vertices are the nonzero
zero-divisors of R. This graph turns out to best exhibit the properties
of the set of zero-divisors of R, and the ideas and problems introduced
in [3] were further studied in [15], [9] and [10]. Let R be a commutative
ring with Z(R) 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, y ∈ R, the vertices x and y are adjacent if
2010 MSC: 16Y60, 05C75.
Key words and phrases: Semiring, Torsion element graph, k-subsemimodules,
QM -subsemimodules.
2 The total torsion element graph of semimodules
and only if x + y ∈ Z(R). The total graph of a commutative ring have
been introduced and studied by D. F. Anderson and A. Badawi in [2].
In [12], the notion of the total torsion element graph of a module over a
commutative ring is introduced.
Subsemimodules of semimodules over semirings play a central role
in the structure theory and are useful for many purposes [13, 14]. How-
ever, they do not in general coincide with the submodules over rings
and, for this reason, their use is somewhat limited in trying to obtain
analogues of ring theorems for semirings. Indeed, many results in rings
apparently have no analogues in semirings using only subsemimodules. In
order to overcome this deficiency, the authors defined a more restricted
class of subsemimodules in semirings, which are called the class of "k-
subsemimodules" and the class of "QM -subsemimodules" [5, 8, 7]. In the
present paper we introduce a new class of graphs, called the total torsion
element graph of a semimodule over a semiring, and we completely char-
acterize the structure of this graph. The total torsion element graph of a
module over a commutative ring and the total torsion element graph of a
semimodule over a commutative semiring are different concepts. Some
of our results are analogous to the results given in [2]. The correspond-
ing results are obtained by modification and here we give a complete
description of the total torsion element graph of a semimodule. The study
of the total torsion element graph of a semimodule M breaks naturally
into two cases depending on whether or not T (M), the set of torsion
elements in M , is a subsemimodule of M . In the third section, we handle
the case when T (M) is a subsemimodule (either k-subsemimodule or
QM -subsemimodule) of M ; in the fourth section, we do the case when
T (M) is not a subsemimodule of M (see Sections 3, 4).
2. Preliminaries
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 a 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, b ∈ V (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 a shortest cycle in Γ, provided Γ contains a cycle;
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 3
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 (respectively,
Γ2) is adjacent (in Γ) to any vertex not in Γ1 (respectively, Γ2).
Throughout this paper R is a commutative semiring with identity. In
order to make this paper easier to follow, we recall in this section various
notions from semimodule theory which will be used in the sequel. For
the definitions of monoid, semirings, semimodules and subsemimodules
we refer [13, 14, 7, 8]. All semiring in this paper are commutative with
non-zero identity. Let M be a semimodule over a semiring R.
(1) A semiring R is said to be semidomain whenever a, b ∈ R with
ab = 0 implies that either a = 0 or b = 0.
(2) A subtractive subsemimodule (= k-subsemimodule) N is a sub-
semimodule of M such that if x, x + y ∈ N , then y ∈ N (so {0M } is a
k-subsemimodule of M).
(3) An element x of M is called a zero-sum in M if x + y = 0 for some
y ∈ M . We use S(M) to denote the set of all zero-sum elements of M .
(4) A semimodule M over a semiring R is called a M -cancellative
semimodule if whenever rm = rn for elements m, n ∈ M and r ∈ R, then
n = m.
(5) A subsemimodule N of a semimodule M over a semiring R is
called a partitioning subsemimodule (= QM -subsemimodule) if there
exists a subset QM of M such that M = ∪{q + N : q ∈ QM } and if
q1, q2 ∈ QM then (q1 + N) ∩ (q2 + N) 6= ∅ if and only if q1 = q2. Let N
be a QM -subsemimodule of M and let M/N = {q + N : q ∈ QM }. Then
M/N forms an R-semimodule under the operations ⊕ and ⊙ defined as
follows: (q1 + N) ⊕ (q2 + N) = q3 + N , where q3 ∈ QM is the unique
element such that q1 + q2 + N ⊆ q3 + N and r ⊙ (q1 + N) = q4 + I, where
r ∈ R and q4 ∈ QM is the unique element such that rq1 + N ⊆ q4 + N .
This R-semimodule M/N is called the quotient semimodule of M by N [7].
By [7, Lemma 2.3], there exists a unique element q0 ∈ QM such that
q0 + N = N . Thus q0 + N is the zero element of M/N .
(6) A torsion element is an element m ∈ M for which there exists a
non-zero element r of R such that rm = 0. The set of torsion elements
in M will be denoted by T (M). Also, we use T (M)∗ to denote the set of
non-zero torsion elements of M .
(7) We define the total torsion element graph of a semimodule M ,
denoted by T (Γ(M)), as follows: V (T (Γ(M))) = M , E(T (Γ(M))) =
4 The total torsion element graph of semimodules
{{x, y} : x + y ∈ T (M)}. We will use Tof(M) to denote the set of
elements of M that are not torsion elements. Let Tof(Γ(M)) be the
(induced) subgraph of T (Γ(M)) with vertices Tof(M), and let Tor(Γ(M))
be the (induced) subgraph of T (Γ(M)) with vertices T (M).
3. T (M) is a subsemimodule of M
Let M be a semimodule over a semiring R. The structure of the
torsion element graph T (Γ(M)) may be completely described in those
cases when torsion elements form a subsemimodule. We begin this section
with the extreme cases T (M) = M and T (M) = {0}.
Theorem 3.1. Let M be a semimodule over a semiring R.
(i) T (Γ(M)) is complete if and only if T (M) = M .
(ii) T (Γ(M)) is totally disconnected if and only if T (M) = S(M) =
{0}.
Proof. (i) If T (M) = M , then for any two vertices x, y ∈ M , one has
x + y ∈ T (M); hence they are adjacent in T (Γ(M)). Conversely, assume
that T (Γ(M)) is complete and let m ∈ M . Then m is adjacent to 0. Thus
m = m + 0 ∈ T (M), and hence we have equality.
(ii) Let T (Γ(M)) be totally disconnected. Then 0 is not adjacent to
any vertex; hence x = x + 0 /∈ T (M) for every non-zero element x of
M . Thus T (M) = {0}. If there is a non-zero element m of S(M), then
there exists 0 6= m′ ∈ M such that m + m′ = 0 ∈ T (M), which is a
contradiction. Thus S(M) = {0}. Conversely, assume that there exist
distinct a, b ∈ M such that a + b ∈ T (M) = {0}. Then a, b ∈ S(M), a
contradiction. Hence T (Γ(M)) is totally disconnected.
Proposition 3.2. Let M be a semimodule over a commutative semir-
ing R.
(i) If 2 = 1R + 1R ∈ Z(R) and x ∈ M , then 2x ∈ T (M). In particular,
if T (M) = {0}, then M is an R-module.
(ii) If x ∈ Tof(M), then 2 ∈ Z(R) if and only if 2x ∈ T (M).
Proof. (i) By assumption, there exists 0 6= r ∈ R such that 2r = 0. Since
r(2x) = (2r)x = 0, we have 2x ∈ T (M). Finally, by assumption, 2x = 0
for every x ∈ M , as required.
(ii) By (i), it is enough to show that if 2x ∈ T (M), then 2 ∈ Z(R).
There exists a non-zero element s of R such that (2s)x = s(2x) = 0; hence
2s = 0 since x /∈ T (M). Thus 2 ∈ Z(R).
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 5
Example 3.3. (1) A subsemimodule of a semimodule over a semiring
in general need not be a k-subsemimodule and QM -subsemimodule. Let
M = R be the set of all real numbers x satisfying 0 < x ≤ 1, and define
a + b = a.b = min{a, b} for all a, b ∈ R. Then (R, +, .) is easily checked to
be a commutative semiring with 1 as identity. Each real number r such
that 0 < r < 1 defines a subsemimodule N = {y ∈ M : y ≤ r} of M .
However, r + 1 = 1 together r ∈ N and 1 /∈ N show that N is not a k-
subsemimodule of M . In particular, N is not a QM -subsemimodule of M
since every QM -subsemimodule is a k-subsemimodule by [7, Theorem 3.2].
(2) Let R = M denote the semiring of nonnegative integers with the
usual operations of addition and multiplication. If m ∈ M − {0}, the
subsemimodule
N = {km : k ∈ R}
is a QM -subsemimodule of M when QM = {0, 1, · · · , m−1}. In particular,
N is a k-subsemimodule.
(3) Assume that R is the semiring of nonnegative integers with the
usual operations of addition and multiplication and let M = (R, gcd). It
is easy to see that M is a commutative monoid in which every element is
idempotent. Hence M is an R-semimodule in which N = {0, 2, 4, · · · } is
a k-subsemomodule of M but not a QM -subsemimodule.
Proposition 3.4. Let M be a semimodule over a semidomain R.
(i) T (M) is a k-subsemimodule of M .
(ii) If M is a M -cancellative semimodule, then T (M) is a QM -sub-
semimodule of M .
Proof. (i) Clearly, T (M) is a subsemimodule of M . Let x + y, x ∈ T (M)
for some x, y ∈ M . Then rx = s(x + y) = 0 for some 0 6= r, s ∈ R (so
rs 6= 0). Therefore, (rs)y = 0, and so y ∈ T (M).
(ii) Set QM = (M − T (M)) ∪ {0}; we show that T (M) is a QM -
subsemimodule of M . Let m ∈ M . If m ∈ T (M), then m ∈ 0M + T (M).
If m /∈ T (M), then m ∈ m + T (M). So M = ∪{q + T (M) : q ∈ QM }. Let
(q1 + T (M)) ∩ (q2 + T (M)) 6= ∅. So q1 + a = q2 + b for some a, b ∈ T (M).
It follows that ra = sb = 0 for some 0 6= r, s ∈ R. Therefore, rsq1 = rsq2
with rs 6= 0 since R is a semidomain. Thus q1 = q2 since M is a M -
cancellative semimodule, as needed.
6 The total torsion element graph of semimodules
Proposition 3.5. Let M be a semimodule over a commutative semiring
R such that T (M) is a subsemimodule of M .
(i) Tor(Γ(M)) is a complete (induced) subgraph of T (Γ(M)).
(ii) If N is a subsimimodule of M , then T (Γ(N)) is the induced
subgraph of T (Γ(M)) if and only T (N) = N ∩ T (M).
(iii) If (0 : M) 6= 0, then T (Γ(M)) is a complete graph.
Proof. The proofs are straightforward.
Example 3.6 shows that there are some semimodules over commutative
semirings such that their torsion subsemimodules are not k-subsemimodues.
Example 3.6. Assume that E+ be the set of all non-negative inte-
gers and let M = R = {(a, b) : a, b ∈ E+}. Define (a, b) + (c, d) =
(min{a, c}, max{b, d}) and (a, b) ∗ (c, d) = (ac, bd) for all (a, b), (c, d) ∈ R.
Then (R, +, ∗) is easily checked to be a commutative semiring. An in-
spection will show that T (M) = {(a, b) ∈ M : a = 0 or b = 0} is
a subsemimodule of M . However, (0, 1) + (2, 5) = (0, 5) ∈ T (M) to-
gether with (2, 5) /∈ T (M) and (0, 1) ∈ T (M) show that T (M) is not a
k-subsemimodule of M . Also, T (Γ(M)) is a connected graph since every
element is adjacent to (0, 0) in T (Γ(M)). Moreover, gr(T (Γ(M))) = 3
since there is a 3-cyclic (0, 0) − (0, 1) − (1, 0) − (0, 0) in T (Γ(M)).
Example 3.7 shows that there are some semimodules over commutative
semirings such that their torsion subsemimodules are k-subsemimodues
but they are not QM -subsemimodules.
Example 3.7. Assume that R = M is the set of all non-negative integers
and let a, b, k ∈ R. Define a + b = gcd(a, b) and
a ∗ b =
0 if gcd(a, b) = 2k,
1 if gcd(a, b) = 2k + 1,
0 a = 0 or b = 0.
Then (R, +, ∗) is easily checked to be a commutative semiring. An in-
spection will show that T (M) = {0, 2, 4, 6, · · · } is a k-subsemimodule of
M but is not a QM -subsemimodule of M by Example 3.3 (3). Moreover,
Tor(Γ(M)) is a complete graph and Tof(Γ(M)) is a totally disconnected
graph.
The main goal of this section is a general structure theorem (The-
orem 3.10) for Tof(Γ(M)) when either T (M) is a k- subsemimodule
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 7
of M or T (M) is a QM -subsemimodule. But first, we record the ba-
sic observation that if T (M) is a k-subsemimodule of (resp. T (M) is
not a k-subsemimodule), then the subgraph Tor(Γ(M)) is disjoint from
Tof(Γ(M)) (resp. Tor(Γ(M)) is not disjoint from Tof(Γ(M)). Thus we
will concentrate on the subgraph Tof(Γ(M)) throughout this section.
Theorem 3.8. Let M be a semimodule over a commutative semiring R
such that T (M) is a k-subsemimodule of M . If m and m′ are distinct
elements of Tof(M) that are connected by a path with m + m′ /∈ T (M)
(i.e., if m and m′ are not adjacent), then there is a path in Tof(Γ(M)) of
length at most 2 between m and m′.
Proof. Let T (M) be a k-subsemimodule of M . It is enough to show that
if m1, m2, m3 and m4 are distinct vertices of Tof(M) and there is a path
m1 −m2 −m3 −m4 from m1 to m4, then m1 and m4 are adjacent. Now we
have m1 + m2 + m3 + m4 ∈ T (M). Then T (M) being k-subsemimodule of
M gives m1+m4 ∈ T (M), and so m1 and m4 are adjacent, as required.
Compare the next theorem with [12, Theorem 2.1 (1)].
Theorem 3.9. Let M be a semimodule over a commutative semiring R.
(i) If T (M) is a k-subsemimodule of M , then Tor(Γ(M)) is disjoint
from Tof(Γ(M)).
(ii) If T (M) is not a k-subsemimodule of M , then Tor(Γ(M)) is not
disjoint from Tof(Γ(M)).
Proof. (i) Let T (M) is a k-subsemimodule of M . If Tor(Γ(M)) is not dis-
joint from Tof(Γ(M)), then there exist a ∈ T (M) and b ∈ Tof(M) such
that a + b ∈ T (M). Thus b ∈ T (M) since T (M) is a k-subsemimodule of
M which is a contradiction. Thus Tor(Γ(M)) is disjoint from Tof(Γ(M)).
(ii) Assume that T (M) is not a k-subsemimodule of M . So there exist
a ∈ T (M) and b ∈ Tof(M) such that a + b ∈ T (M). Let x ∈ M . We
define the subset PT (x) as follows:
PT (x) = {m ∈ T (M) : there is a path of finite length between x and m}.
Clearly, if x ∈ T (M), then T (M) ⊆ PT (x), and so PT (x) 6= ∅. Set N =
{x ∈ M : PT (x) 6= ∅}. Therefore T (M) ⊂ N , since b ∈ N and b /∈ T (M).
Now, we show that N is a subsemimodule of M . Let x1, y1 ∈ N . Therefore,
there exist m1, m′
1 ∈ T (M), x1, x2, · · · , xn ∈ M and y1, y2, · · · , yk ∈ M
such that x1−x2−· · ·−xn−m1 and y1−y2−· · ·−yk−m′
1 are paths of finite
lengths between x1, m1 and y1, m′
1, and so we have xi+xi+1, yj +yj+1, xn+
8 The total torsion element graph of semimodules
m1, yk + m′
1, m1 + m′
1 ∈ T (M) for each 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ k − 1.
We may assume that n ≤ k. So (xi + yi) + (xi+1 + yi+1) ∈ T (M) for each
1 ≤ i ≤ n − 1 and 1 ≤ j ≤ k − 1. Then
(x1 + y1) − (x2 + y2) − · · · − (xn + yn)−
(m1 + yn+1) − (m′
1 + yn+2) − (m1 + yn+3) − · · · − m1
is a path of finite length between x1 + y1 and m1. Hence PT (x1 + y1) 6= ∅,
and so x1 +y1 ∈ N . Now, let r ∈ R. Therefore, rx1 −rx2 −· · ·−rxn −rm1
is a path between rx1 and rm1 of finite length, and so PT (rx) 6= ∅. Thus
N is a subsemimodule of M and T (M) ⊂ N . It is easy to see that
T (Γ(N)) is a connected subgraph of T (Γ(M)) containing Tor(Γ(M)).
Hence, Tor(Γ(M)) is not disjiont from Tof(Γ(M)).
Compare the next theorem with [12, Theorem 2.5].
Theorem 3.10. Let M be a semimodule over a commutative semiring
R, |T (M)| = α and |QM − T (M)| = β.
(i) If T (M) is a k-subsemimodule of M and 2 ∈ Z(R), then Tof(Γ(M))
is the union of disjoint complete subgraphs.
(ii) If T (M) is a k-subsemimodule of M and 2 /∈ Z(R), then Tof(Γ(M))
is the union of totally disconnected subgraphs and some connected sub-
graphs.
(iii) If T (M) is a QM -subsemimodule of M and 2 ∈ Z(R), then
Tof(Γ(M)) is the union of β disjoint Kλ’s such that λ ≤ α.
(iv) If T (M) is a QM -subsemimodule of M and 2 /∈ Z(R), then
Tof(Γ(M)) is the union of totally disconnected subgraphs and complete
bipartite subgraphs.
Proof. (i) Let 2 ∈ Z(R). We set up an equivalence relation ∼ on Tof(M)
as follows: for y, y′ ∈ Tof(M), we write y ∼ y′ if and only if y + y′ ∈
T (M). By Proposition 3.2 and T (M) being a k-subsemimodule of M , it is
straightforward to check that ∼ is an equivalence relation on Tof(M): for
y ∈ Tof(M), we denote the equivalence class which contains y by [y]. Now
let y ∈ Tof(M). If [y] = {y}, then (y + a) + (y + b) = 2y + (a + b) ∈ T (M)
for every a, b ∈ T (M) by Proposition 3.2. So y + T (M) is a complete
subgraph with at most α vertices. If |[y]| = γ > 1, then for every y′ ∈ [y]
we have (y + a) + (y′ + b) = (y + y′) + a + b ∈ T (M), where a, b ∈ T (M).
Thus y + T (M) is a part of a complete graph Kν with ν ≤ αγ vertices.
Therefore, Tof(Γ(M)) is the union of disjoint complete subgraphs.
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 9
(ii) Let 2 /∈ Z(R) and y ∈ Tof(M). Set
Ay = {y′ ∈ Tof(M) : y + y′ ∈ T (M)}.
If Ay = ∅, then y + y′ /∈ T (M) for every y′ ∈ Tof(M). In this case, we
show that y + T (M) is a totally disconnected subgraph of Tof(Γ(M)).
If (y + a) + (y + b) ∈ T (M) for some a, b ∈ T (M), then 2y + a + b =
(y + a) + (y + b) ∈ T (M); so 2y ∈ T (M), which is a contradiction by
Proposition 3.2. Therefore, y+T (M) is a totally disconnected subgraph of
Tof(Γ(M)). We may assume that Ay 6= ∅. Then y + y′ ∈ T (M) for some
y′ ∈ Tof(M). Thus (y +a)+(y′ + b) = (y +y′)+(a+ b) ∈ T (M) for every
a, b ∈ T (M); hence each element of y+T (M) is adjacent to each element of
y′ + T (M). If |Ay| = ν, then we have a connected subgraph of Tof(Γ(M))
with at most αν vertices. Hence, If 2 /∈ Z(R), then Tof(Γ(M)) is the
union of totally disconnected subgraphs and some connected subgraphs.
(iii) First, we show that q+T (M) ⊆ Tof(M) for every q ∈ QM −T (M).
If q + a /∈ Tof(M) for some a ∈ T (M), then q + a ∈ T (M); hence
q ∈ T (M) since T (M) is a k-subsemimodule which is a contradiction.
Let 2 ∈ Z(R) and q ∈ QM − T (M). Then each coset q + T (M) is a
complete subgraph of Tof(M) with λ vertices such that λ ≤ α (note that
(q1+T (M))∩(q2+T (M)) 6= ∅ if and only if q1 = q2) since (q+a)+(q+b) =
2q + (a + b) ∈ T (M) for all a, b ∈ T (M) by Proposition 3.2 and T (M)
is a subsemimodule. Now we show that distinct cosets form disjoint
subgraphs of Tof(Γ(M)). If q1 + a and q2 + b are adjacent for some
q1, q2 ∈ QM − T (M) and a, b ∈ T (M), then (q1 + a) + (q2 + b) ∈ T (M)
gives q1+q2 ∈ T (M) since T (M) is a k-subsemimodule of M . So q2+2q1 =
q1 +(q1 + q2) ∈ q1 +T (M). Likewise, q2 +2q1 ∈ q2 +T (M) by Proposition
3.2. So q2 + 2q1 ∈ (q1 + T (M)) ∩ (q2 + T (M)); hence q1 = q2. Thus
Tof(Γ(M)) is the union of β disjoint induced subgraphs q + T (M), each
of which is a Kλ such that λ ≤ α.
(iv) Next assume that 2 /∈ Z(R) and let q ∈ QM − T (M). If q + q′ /∈
T (M) for every q′ ∈ QM − T (M), then Aq = ∅. Then by (ii), q + T (M) is
a totally disconnected subgraph of Tof(Γ(M)). So we may assume that
q + q′ ∈ T (M) for some q′ ∈ QM − T (M). Then by (ii) each element
of q + T (M) is adjacent to each element of q′ + T (M); we show that q′
is the unique element. Let q + q′′ ∈ T (M) for some q′′ ∈ QM − T (M).
Therefore, q + q′ + q′′ = q′ + (q + q′′) ∈ q′ + T (M). Likewise, q + q′ + q′′ =
q′′+(q+q′) ∈ q′′+T (M). Thus (q′+T (M))∩(q′′+T (M)) 6= ∅ gives q′ = q′′.
Therefore (q + T (M)) ∪ (q′ + T (M)) is a complete bipartite subgraph of
Tof(Γ(M)). So Tof(Γ(M)) is the union of totally disconnected subgraphs
and complete bipartite subgraphs.
10 The total torsion element graph of semimodules
Proposition 3.11. Let M be a semimodule over a commutative semir-
ing R.
(i) If T (M) is a k-subsemimodule of M and Tof(Γ(M)) is complete,
then |Tof(M)| = 1 or |Tof(M)| = 2.
(ii) If T (M) is a QM -subsemimodule of M and Tof(Γ(M)) is complete,
then |M/T (M)| = 2 or |M/T (M)| = 3.
(iii) If T (M) is a QM -subsemimodule of M , |M/T (M)| = 2 and
2 ∈ Z(R), then Tof(Γ(M)) is complete.
Proof. (i) By Proposition 3.2, 2y ∈ T (M) for every y ∈ Tof(M). Then
y + T (M) is a complete subgraph of Tof(Γ(M)); hence |Tof(M)| = 1
since Tof(Γ(M)) is complete. If 2 /∈ Z(R), then for each y ∈ Tof(M),
there exists y′ ∈ Tof(M) such that y + y′ ∈ T (M). So |Tof(M)| = 2
since Tof(Γ(M)) is complete. In this case, Tof(Γ(M)) is a complete
bipartite graph (see Theorem 3.10).
(ii) Since every QM -subsemimodule is a k-subsemimodule, the part
(i) gives |Tof(M)| = 1 or |Tof(M)| = 2. If |Tof(M)| = 1, then M =
T (M) ∪ (q + T (M)) for q ∈ Tof(M) and hence |M/T (M)| = 2. SimilarIy,
if |Tof(M)| = 2, then M = T (M) ∪ (q + T (M)) ∪ (q′ + T (M)) for
q, q′ ∈ Tof(M) with q 6= q′, and hence |M/T (M)| = 3.
(iii) Let |M/T (M)| = 2 and 2 ∈ Z(R). Then M = T (M)∪ (q +T (M))
for some q ∈ QM − T (M); so 2q ∈ T (M) by Proposition 3.2. Let m, m′ ∈
Tof(M). Then m, m′ ∈ q + T (M). So m + m′ = (q + a) + (q + b) = 2q +
(a + b) ∈ T (M) for some a, b ∈ T (M). Thus Tof(Γ(M)) is complete.
Proposition 3.12. Let M be a semimodule over a commutative semiring
R such that T (M) is a QM -subsemimodule of M . Then:
(i) If Tof(Γ(M)) is connected, then |M/T (M)| = 2 or |M/T (M)| = 3.
(ii) If |M/T (M)| = 2 and 2 ∈ Z(R), then Tof(Γ(M)) is connected.
Proof. (i) Let Tof(Γ(M)) be a connected graph. Then Tof(Γ(M)) is a
single complete graph Kλ or a bipartite graph by Theorem 3.10. Hence
Tof(Γ(M)) is a complete graph. Now the assertion follows from Proposi-
tion 3.11.
(ii) This follows directly from Proposition 3.11.
Theorem 3.13. Let M be a semimodule over a commutative semiring R.
(i) If T (M) is a k-subsemimodule of M , then diam(Tof(Γ(M))) = 0
if and only if T (M) = {0} and |M | = 2.
(ii) Let T (M) be a QM -subsemimodule of M . Then the following
hold:
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 11
(a) diam(Tof(Γ(M))) = 1 if and only if 2 ∈ Z(R) and |M/T (M)| = 2.
(b) diam(Tof(Γ(M))) = 2 if and only if |M/T (M)| = 3, 2 /∈ Z(R)
and q + q′ ∈ T (M) for every q, q′ ∈ QM − T (M).
(c) Otherwise diam(Tof(Γ(M))) = ∞.
Proof. (i) If diam(Tof(Γ(M))) = 0, then Tof(Γ(M)) is a complete graph
K1, and so |T (M)| = |Tof(M)| = 1 by Theorem 3.10. Hence T (M) = {0}
and |M | = 2. The other implication is clear.
(ii) (a) If diam(Tof(Γ(M))) = 1, then Tof(Γ(M)) is a complete
graph Kλ with λ ≤ |T (M)| by Theorem 3.10. Therefore, 2 ∈ Z(R) and
|QM −T (M)| = 1. Thus M = T (M)∪(q+T (M)) for some q ∈ QM −T (M);
hence |M/T (M)| = 2. The converse follows from Theorem 3.10.
(ii) (b) If diam(Tof(Γ(M))) = 2, then Tof(Γ(M)) is a complete
bipartite graph K1,2 or K2,2; thus 2 /∈ Z(R) and |QM − T (M)| = 2
by Theorem 3.10. Since Tof(Γ(M)) has not any totally disconnected
subgraph, we must have q + q′ ∈ T (M) for every q, q′ ∈ QM − T (M).
Remark 3.14. Let R and M be as described in Example 3.7. So T (M) =
{0, 2, 4, 6, · · · } is a k-subsemimodule of M but is not a Q-subsemimodule
of M . Also, Tor(Γ(M)) is a complete graph and Tof(Γ(M)) is a totally
disconnected graph. Since gcd(2, 4) = 2, we have 2∗4 = 0; hence 2 ∈ Z(R).
Moreover, M = T (M) ∪ (1 + T (M)) and diam(Tof(Γ(M))) = ∞. Hence
Theorem 3.13 (ii) is not true when T (M) is not a QM -subsemimodule
of M .
Proposition 3.15. Let M be a semimodule over a commutative semir-
ing R such that T (M) is a k-subsemimodule of M . Then gr(Tof(Γ(M))) =
3, 4 or ∞. In particular, if Tof(Γ(M)) contains a cycle, gr(Tof(Γ(M))) ≤ 4.
Proof. Let Tof(Γ(M)) contains a cycle. Then Tof(Γ(M)) is not a totally
disconnected graph, so by the proof of Theorem 3.10, Tof(Γ(M)) has
either a complete or a complete bipartite subgraph. Therefore, it must
contain either a 3-cycle or a 4-cycle. Thus gr(Tof(Γ(M))) ≤ 4.
Theorem 3.16. Let M be a semimodule over a commutative semiring
R such that T (M) be a k-subsemimodule of M .
(i) gr(Tof(Γ(M))) = 3 if and only if 2 ∈ Z(R) and |y + T (M)| ≥ 3
for some y ∈ Tof(M).
(ii) gr(Tof(Γ(M))) = 4 if and only if 2 /∈ Z(R) and y + y′ ∈ T (M)
for some y, y′ ∈ Tof(M).
12 The total torsion element graph of semimodules
Proof. (i) Assume that gr(Tof(Γ(M))) = 3. Then by Theorem 3.10,
Tof(Γ(M)) is a complete graph Kλ with 3 ≤ λ. Therefore, 2 ∈ Z(R) and
|y + T (M)| ≥ 3 for some y ∈ Tof(M).
(ii) If gr(Tof(Γ(M))) = 4, then by Theorem 3.10, Tof(Γ(M)) has
a complete bipartite subgraph; hence 2 /∈ Z(R) and y + y′ ∈ T (M) for
some y, y′ ∈ Tof(M) by Theorem 3.10. The other implications of (i) and
(ii) follows directly from Theorem 3.10.
Theorem 3.17. Let M be a semimodule over a commutative semiring
R such that T (M) be a k-subsemimodule of M .
(i) gr(T (Γ(M))) = 3 if and only if |T (M)| ≥ 3.
(ii) gr(T (Γ(M))) = 4 if and only if 2 /∈ Z(R), |T (M)| < 3 and
y + y′ ∈ T (M) for some y, y′ ∈ Tof(M).
(iii) Otherwise, gr(T (Γ(M))) = ∞.
Proof. (i) This follows from Proposition 3.5.
(ii) Since gr(Tor(Γ(M)) = 3 or ∞, then gr(Tof(Γ(M))) = 4. There-
fore, 2 /∈ Z(R) and y + y′ ∈ T (M) for some y, y′ ∈ Tof(M) by Theorem
3.16. On the other hand, gr(T (Γ(M)) 6= 3; so |T (M)| < 3. The converse
implication follows from Theorem 3.10.
4. T (M) is not a subsemimodule of M
We continue to use the notation already established, so M is a semi-
module over a commutative semiring R. In this section , we study the
torsion element graph T (Γ(M)) when T (M) is not a subsemimodule of M .
Lemma 4.1. Let M be a semimodule over a semiring R such that T (M)
is not a subsemimodule of M . Then there are distinct m, m′ ∈ T (M)∗
such that m + m′ ∈ Tof(M).
Proof. It is enough to show that T (M) is always closed under scalar
multiplication of its elements by elements of R. Let m ∈ T (M) and r ∈ R.
There is a non-zero element s ∈ R with sm = 0; hence s(rm)) = r(sm) =
0. Thus rm ∈ T (M). This completes the proof.
Theorem 4.2. Let M be a semimodule over a semiring R such that
T (M) is not a subsemimodule of M . Then Tor(Γ(M)) is connected with
diam(Tor(Γ(M))) = 2.
Proof. Let x ∈ T (M)∗. Then x is adjacent to 0. Thus x − 0 − y is a
path in Tor(Γ(M)) of length two between any two distinct x, y ∈ T (M)∗.
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 13
Moreover, there exist nonadjacent x, y ∈ T (M)∗ by Lemma 4.1; thus
diam(Tor(Γ(M))) = 2.
Example 4.3. Let R = {0, 1, a} be the idempotent semiring in which
1 + a = a + 1 = a and let M = R ⊕ R. Then M is a semimod-
ule over R with 9 elements. An inspection will show that T (M) =
{(0, 0), (1, 0), (0, 1), (a, 0), (0, a)} is not a subsemimodule of M and M =
〈T (M)〉. Moreover, Tor(Γ(M)) is disjoint from Tof(Γ(M)) and Tof(Γ(M))
is a totally disconnected subgraph of T (Γ(M)). Hence T (Γ(M)) is dis-
connected. So Theorem 3.1 (ii), (iii) and Theorem 3.2 in [12], in general,
are not true when M is a semimodule over a semiring R.
Definition 4.4. A semimodule M over a semiring R is called a subtractive
semimodule if every cyclic subsemimodule of M is a k-subsemimodule.
Example 4.5. Assume that E+ be the set of all non-negative integers
and let M = R = E+ ∪{∞}. Define a+b = max{a, b} and ab = min{a, b}
for all a, b ∈ R. Then R is a commutative semiring with 1R = ∞ and
0R = 0. An inspection will show that the list of subsemimodules of M
are: M , E+ and for every non-negative integer n
Nn = {0, 1, ..., n}.
It is clear that every proper subsemimodule of M is a k-subsemimodule.
So M is a subtractive semimodule.
Lemma 4.6. Let R be a semiring which is not a ring, and let M be a
subtractive R-semimodule. Then S(M) ⊆ T (M).
Proof. If S(M) = {0}, we are done. Suppose that 0 6= x ∈ S(M). Then
there is a y ∈ S(M) such that x + y = 0. Thus y ∈ Rx since Rx is a
k-subsemimodule. Then there exists r ∈ R such that (1 + r)x = 0. It
then follows from [11, Lemma 2.1] that 1 + r 6= 0. Thus x ∈ T (M), as
required.
Theorem 4.7. Let R be a semiring which is not a ring, and let M be a
subtractive R-semimodule. If |S(M)| ≥ 3, then gr(Tor(Γ(M))) = 3.
Proof. Let 0 6= x, y ∈ S(M). Then x, y ∈ T (M) by Lemma 4.6 and
x + y = 0 ∈ T (M). Thus 0 − x − y − 0 is a 3-cycle in Tor(Γ(M)).
Theorem 4.8. Let M be a semimodule over a commutative semiring R
such that T (M) is not a subsemimodule of M .
(i) If T (Γ(R)) is connected, then T (Γ(M)) is connected.
(ii) Either gr(Tor(Γ(M))) = 3 or gr(Tor(Γ(M))) = ∞.
14 The total torsion element graph of semimodules
Proof. (i) It is clear that if r ∈ Z(R) and x ∈ M , then rx ∈ T (M).
Assume that y ∈ M and let 0 − a1 − a2 − · · · − ak − 1 be a path from
0 to 1 in T (Γ(R)). Then 0 − a1y − a2y − · · · − aky − y is a path from 0
to y in T (Γ(M)). Since all vertices may be connected via 0, T (Γ(M)) is
connected.
(ii) If x + y ∈ T (M) for some distinct x, y ∈ T (M)∗, then 0 − x − y − 0
is a 3-cycle in Tor(Γ(M)); so gr(Tor(Γ(M))) = 3. Otherwise, x + y ∈
Tof(M) for all distinct x, y ∈ T (M) by Lemma 4.1. Therefore, in this
case, each x ∈ T (M)∗ is adjacent to 0, and no two distinct x, y ∈ T (M)∗
are adjacent. Thus Tor(Γ(M)) is a star graph with center 0; hence
gr(Tor(Γ(M))) = ∞.
Lemma 4.9. Let M be a semimodule over a commutative semiring R
such that T (M) is not a subsemimodule of M . Then |Z(R)| ≥ 3.
Proof. By Lemma 4.1, there are distinct x, y ∈ T (M) such that x + y ∈
Tof(M). Thus there exist 0 6= r, s ∈ R such that rx = sy = 0. Since
rs(x + y) = 0 and x + y /∈ T (M), we have rs = 0, as needed.
Theorem 4.10. Let M be a semimodule over a commutative semiring
R such that T (M) is not a subsemimodule of M .
(i) If Z(R) is an ideal of R, then gr(Tor(Γ(M))) = 3
(ii) If Z(R) is not an ideal of R, then gr(Tof(Γ(M))) = 3 or ∞.
Proof. (i) Let Z(R) is an ideal of R. By Lemma 4.9, there are non-
zero elements r, s of Z(R) with r + s ∈ Z(R) and rs = 0. Therefore,
t(r + s) = 0 for some non-zero element t of R. Now let m, m′ ∈ Tof(M).
Since t(r + s)m = t(r + s)m′ = 0, we have rm + sm, rm′ + sm′ ∈ T (M).
On the other hand, rm + rm′, sm + sm′ ∈ T (M) since s(rm + rm′) =
r(sm+sm′) = 0. Hence rm−rm′−sm′−sm−rm is a 4-cycle in T (Γ(M)).
Then gr(Tor(Γ(M))) = 3 by Theorem 4.8 (ii).
(ii) We may assume that Tof(Γ(M)) contains a cycle. So there is
a path x − y − z in Tof(M). If x + z ∈ T (M), then we have a 3-cycle
in Tof(Γ(M)). So we may assume that x + z /∈ T (M). There exist
r1, r2 ∈ Z(R) such that r1 +r2 /∈ Z(R) since Z(R) is not an ideal of R. So
there are 0 6= t1, t2 ∈ R such that r1t1 = r2t2 = 0 and then t1t2 = 0 since
t1t2(r1 + r2) = 0. Therefore t1x + t1z ∈ T (M) since t2(t1x + t1z) = 0.
Thus t1x − t1y − t1z − t1x is a 3-cycle in Tof(Γ(M)) and the proof is
complete.
S. Ebrahimi Atani, F. Esmaeili Khalil Saraei 15
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Contact information
S. Ebrahimi Atani,
F. Esmaeili
Khalil Saraei
Faculty of Mathematical Sciences, University of
Guilan, P.O. Box 1914, Rasht, Iran
E-Mail: ebrahimiatani@gmail.com
Received by the editors: 01.09.2011
and in final form 26.04.2012.
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