Journal ArticleDOI

# Domination number in the annihilating-ideal graphs of commutative rings

01 Jan 2015-Publications De L'institut Mathematique (National Library of Serbia)-Vol. 97, Iss: 111, pp 225-231
TL;DR: In this paper, the annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(r)\{0} and two distinct vertices I====== and J are adjacent if and only if IJ = 0.
Abstract: Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

### 1. Introduction

• The study of algebraic structures, using the properties of graphs, have become an exciting research topic in the past twenty years, leading to many interesting results and questions.
• Throughout this paper, all rings are assumed to be commutative rings with identity.
• A subset 2010 Mathematics Subject Classification: Primary 13A15; Secondary 05C75.
• Also, some relations between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are studied.

### 2. Main results

• The authors start with the following remark which completely characterizes all rings for which either γ(AG(R)) = 1 or γ(Γ(R)) = 1. Remark 2.1.
• Now, the authors can state the following proposition.
• The following theorem provides an upper bound for the domination number of the annihilating-ideal graph of a Noetherian ring.

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PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 97(111) (2015), 225–231 DOI: 10.2298/PIM140222001N
DOMINATION NUMBER
IN THE ANNIHILATING-IDEAL GRAPHS
OF COMMUTATIVE RINGS
Reza Nikandish, Hamid Reza Maimani, and Sima Kiani
Abstract. Let R be a commutative ring with identity and A(R) be the set of
ideals with nonzero annihilator. The annihilating-ideal graph of R is deﬁned
as the graph AG(R) with the vertex set A(R)
= A(R) r {0} and two distinct
vertices I and J are adjacent if and only if I J = 0. In this paper, we study the
domination number of AG(R) and some connections between the domination
numbers of annihilating-ideal graphs and zero-divisor graphs are given.
1. Introduction
The study of algebraic structures, using the properties of graphs, have become
an exciting research topic in the past twenty years, leading to many interesting
results and questions. There are many papers on assigning a graph to a ring, for
instance see [1, 3, 9, 11].
Throughout this paper , all r ings are assumed to be commutative rings with
identity. By Min(R), Z(R) and Nil(R) we denote the set o f all minimal prime
ideals of R, the s e t of all zero-divisors of R and the set of all nilpotent elements of
R, respectively. The s ocle of ring R, denoted by Soc(R), is the sum of all minimal
ideals of R. If there are no minimal ideals, this sum is deﬁned to be zero. A prime
ideal p is said to be an associated prime ideal of a commutative Noetherian r ing
R, if there exists a nonzero element x in R such that p = ann(x). By Ass(R) we
denote the set of all associated prime ideals of R. A ring R is said to be reduced, if
it has no nonzero nilpotent element or equivalently
T
P Min(R)
P = 0.
For every graph G, we denote by V (G), the vertex set of G. A bipartite graph
is a graph all of whose vertices can be partitioned into two parts U and V such that
every edge joins a vertex in U to one in V . A complete bipartite graph is a bipartite
graph in which every vertex of one part is joined to every vertex of the other part.
If o ne of the parts is a singleton, then the graph is said to be a star graph. A subset
2010 Mathematics Subject Classiﬁcation: Primary 13A15; Secondary 05C75.
Key words and phrases: annihilating-ideal graph, zero-divisor graph, domination number,
minimal prime ideal.
The research of H. R. Maimani was partially supported by the grant 91050214 from IPM.
Communicated by Žarko Mijajlović.
225

226 NIKANDISH, MAIMANI, AND KIANI
D of V (G) is called a dominating set if every vertex of G is either in D or adjacent
to at least one vertex in D. The domination number of G, denoted by γ(G), is
the number of vertices in a smallest dominating set of G. A total dominating s et
of a graph G is a set S of vertices of G such that every vertex is adjacent to a
vertex in S. The total domination number of G, denoted by γ
t
(G), is the minimum
cardinality of a total dominating set. A dominating set of ca rdinality γ(G) (γ
t
(G))
is called a γ-set (γ
t
-set).
Let R be a ring. The zero-divisor graph of R, Γ(R ), is a graph with the vertex
set Z(R) r {0} and two distinct vertices x and y are adjacent if and only if xy = 0.
The concept of the zero-divisor graph was ﬁrst introduced by Be ck (see [5]). We
call an idea l I of R, an annihilating-ideal if there exists a nonzero ideal J of R such
that IJ = 0. We use the notation A(R) for the set of a ll annihilating-ideals of R.
By the annihilating-ideal graph of R, AG(R), we mean the graph with the vertex
set A(R)
= A(R) r {0} such that two distinct vertices I a nd J are adjacent if and
only if IJ = 0. The annihilating-ideal graph was ﬁrst introduced in [6] a nd some
interesting properties of this g raph have been studied. In this article, we study
the domination number of the annihilating-ideal graphs. For reduced rings with
ﬁnitely many minimal primes and Artinian rings , the domination numbe r of the
annihilating-ideal graphs is given. Also, s ome relations between the domination
numbers of annihilating-ideal g raphs and zero-divisor graphs are studied.
2. Main results
We start with the following remark which completely character ize s all rings for
which either γ(AG(R)) = 1 or γ(Γ(R)) = 1.
Remark 2.1. Let R be a ring. By [6, Theorem 2.2], there is a vertex of AG(R)
which is adjacent to every other vertex if a nd only if either R
=
F × D, where F
is a ﬁeld and D is an integral domain, or Z(R) is an annihilator ideal. Also, by
[3, Theorem 2.5], there is a vertex of Γ(R) which is adjacent to every other vertex
if and only if either R
=
Z
2
× D, where D is an integral domain, or Z(R) is an
annihilator ideal. Now, let R be a reduced ring. Then γ(AG(R )) = 1 if and only if
R
=
F × D, where F is a ﬁeld and D is an integral domain. Moreover, γ(Γ(R)) = 1
if and only if R
=
Z
2
× D, wher e D is an integral domain.
Now, we can state the following proposition.
Proposition 2.1. Let R be a ring. Then we have the following:
(i) If γ(AG(R)) = 1, then γ(Γ(R)) { 1, 2}.
(ii) If γ(Γ(R)) = 1, then γ(AG(R)) = 1.
Proof. The result follows from Remark 2.1 and [10, Pr oposition 8].
The following res ult describes the relation b e tween γ
t
(AG(R)) (γ
t
(Γ(R))) and
γ(AG(R)) (γ(Γ(R))).
Theorem 2.1. Let R be a ring. Then
(i) γ
t
(AG(R)) = γ(AG(R)) or γ
t
(AG(R)) = γ(AG(R)) + 1.

DOMINATION NUMBER IN THE ANNIHILATING-IDEAL GRAPHS 227
(ii) γ
t
(Γ(R)) = γ(Γ(R)) or γ
t
(Γ(R)) = γ(Γ(R)) + 1.
Proof. (i) Let γ
t
(AG(R)) 6= γ(AG(R )) and D be a γ-set of AG(R). If
γ(AG(R)) = 1, then it is clear that γ
t
(AG(R)) = 2. So let γ(AG(R)) > 1
and put k = Max
n | I
1
, . . . , I
n
D s.t.
Q
n
i=1
I
i
6= 0
. Since γ
t
(AG(R)) 6=
γ(AG(R)), we have k > 2. Let I
1
, . . . , I
k
D be such that
Q
k
i=1
I
i
6= 0. Then
S = {
Q
k
i=1
I
i
, ann I
1
, . . . , ann I
k
}Dr{I
1
, . . . , I
k
} is a γ
t
-set. Hence γ
t
(AG(R)) =
γ(AG(R)) + 1.
(ii) It is clear by the same argument of part (i).
In the following result we ﬁnd the total domination number of AG(R).
Theorem 2.2. Let M be the set of all maximal elements of the set A(R). If
|M| > 1, then γ
t
(AG(R)) = |M|.
Proof. Let M be the set of all maximal elements of the set A(R), I M and
|M| > 1. First we show that I = ann(ann I) and there exists x R such that
I = ann(x). Let I M. Then ann I 6= 0 and so there exists 0 6= x ann I. Hence
I ann(ann I) ann(x). Thus by the maximality of I, we have I = ann(ann I) =
ann(x). By Zorn’s Lemma it is clear that if A(R) 6= , then M 6= . Fo r any I M
choose x
I
R such that I = ann(x
I
). We assert that D = {Rx
I
| I M } is a total
dominating set of AG(R). Since for every J A(R) there exists I M such that
J I = ann(x
I
), J and Rx
I
are adjacent. Also for each pair I, I
M, we have
Rx
I
Rx
I
= 0. Na mely, if there exists x Rx
I
Rx
I
r {0}, then I = I
= ann(x).
Thus γ
t
(AG(R)) 6 |M |. To complete the proof, we show that each e lement of a n
arbitrary γ
t
-set of AG(R) is adjacent to exactly one element of M. Assume to the
contrary, that a vertex K o f a γ
t
-set of AG(R) is adjacent to I and I
, for I, I
M.
Thus I = I
= ann K, which is imposs ible. Therefo re γ
t
(AG(R)) = |M|.
Theorem 2.3. Let R be a ring. Then γ
t
(Γ(R)) 6 γ
t
(AG(R)).
Proof. Let γ
t
(AG(R)) = k. In the light of the proof of Theorem 2.2, there
exists x
i
R such that ann(x
i
) is a maximal ideal of A(R), for i = 1, . . . , k,
and D = {Rx
1
, . . . , Rx
k
} is a minimum total dominating set of AG(R). It is
easy to check that D
= {x
1
, . . . , x
k
} is a total dominating set of Γ(R). Thus
γ
t
(Γ(R)) 6 γ
t
(AG(R)).
It is interes ting to ﬁnd some rings for which γ
t
(Γ(R)) = γ(Γ(R)) = γ
t
(AG(R))
= γ(AG(R)). In the next result, we study the domination number of the annihi-
lating-ideal graphs of reduced rings with ﬁnitely many minimal primes.
Theorem 2.4. Let R be a reduced ring and | Min(R )| < . If γ(AG(R)) > 1,
then γ(AG(R)) = γ
t
(AG(R)) = | Min(R)|.
Proof. Since R is reduced and γ(AG(R)) > 1, we have | Min(R)| > 1 . Sup-
pose that Min(R) = {p
1
, . . . , p
n
}. If n = 2, the result follows from [7, Corol-
lary 2.5]. Therefore, suppose that n > 3. Deﬁne
b
p
i
= p
1
. . . p
i1
p
i+1
. . . p
n
, for
every i = 1, . . . , n. Clear ly,
b
p
i
6= 0, for every i = 1, . . . , n. Since R is reduced, we
deduce that
b
p
i
p
i
= 0. Therefore, every p
i
is a vertex of AG(R). If I is a vertex of

228 NIKANDISH, MAIMANI, AND KIANI
AG(R), then by [8, Corollary 2.4], I Z(R) =
S
n
i=1
p
i
. It follows from the Prime
Avoidance Theorem (see [12, Theorem 3.61]) that I p
i
, for some i, 1 6 i 6 n.
Thus p
i
is a maximal element of A(R), for every i = 1, . . . , n. From Theorem 2.2,
γ
t
(AG(R)) = | Min(R)|. Now, we show that γ(AG(R)) = n. Assume to the con-
trary, that B = {J
1
, . . . , J
n1
} is a dominating s e t for AG(R). Since n > 3, the
ideals p
i
and p
j
, for i 6= j are no t adjacent (from p
i
p
j
= 0 p
k
it would follow
that p
i
p
k
, or p
j
p
k
which is not true). Because of that, we may assume that
for some k < n 1, J
i
= p
i
for i = 1, k, but none of the other of ideals from B are
equal to some p
s
(if B = {p
1
, . . . , p
n1
} then p
n
would be adjacent to some p
i
, for
i 6= n). So, every ideal in {p
k+1
, . . . , p
n
} is adjacent to an ideal in {J
k+1
, . . . , J
n1
}.
It follows that for some s 6= t there is an l such that p
s
J
l
= 0 = p
t
J
l
. Since p
s
* p
t
,
it follows that J
l
p
t
, so J
2
l
= 0, which is impossible, since the ring R is reduced.
So γ(AG(R)) = γ
t
(AG(R)) = | Min(R)|.
Theorem 2.4 leads to the following corollar y.
Corollary 2.1. Let R be a reduced ring. If γ(AG(R)) > 1, then the following
are equivalent:
(i) γ(AG(R)) = 2.
(ii) AG(R ) is a bipartite graph with two nonempty parts.
(iii) AG(R) is a complete bipartite graph with two n onempty parts.
(iv) R has exactly two minimal primes.
Proof. The result follows from Theorem 2.4 and [7, Corollary 2.5].
In Theorem 15 of [10], it is proved that if R is a ﬁnite reduced ring such that
γ(Γ(R)) 6= 1, then γ(Γ(R)) = | Min(R)|. In the next theorem, we prove this result,
where R is not necessarily ﬁnite.
Theorem 2.5. Let R be a reduced ring and | Min(R)| < . If γ(Γ(R)) > 1,
then γ(Γ(R)) = γ
t
(Γ(R)) = | Min(R)|.
Proof. Using the notatio ns in the proof of Theorem 2.4, set A = {bx
i
| 1 6
i 6 n}, where, for every i, bx
i
is an element of
b
p
i
. We show that A is a dominating
set in Γ(R). Since R is reduced, it is easily seen that bx
i
is a vertex of Γ(R), for
i = 1, . . . , n. Assume that x / A is a vertex of Γ(R). Then x p
i
, for some i. The
equality bx
i
p
i
= 0 implies that xbx
i
= 0. In the sequel, we prove that γ(Γ(R)) = n.
If n = 2, then [2, Theorem 2.4] completes the proof. Thus ass ume that n > 3.
Assume to the contrary, the set B = {y
1
, . . . , y
n1
} is a dominating set for Γ(R).
By the Prime Avoidance Theorem, there exists x
i
p
i
r
S
n
j=1,j6=i
p
j
. Thus there
exists k, 1 6 k 6 n 1, such that y
k
x
i
= y
k
x
j
= 0, for some diﬀerent i, j,
1 6 i, j 6 n. Since x
i
/ p
j
and x
j
/ p
i
, we have y
k
p
i
p
j
. As R is a reduced
ring, we conclude that y
k
/ p
l
, for some l, 1 6 l 6 n. Now, y
k
x
i
= 0 p
l
implies
that e ither y
k
p
l
or x
i
p
l
i
bx
j
= 0, where i 6= j.
Therefore, γ(Γ(R)) = γ
t
(Γ(R)) = | Min(R)|.
Corollary 2.2. Let R be a reduced ring and | Min (R)| < . If γ(AG (R)) 6= 1,
then γ(AG(R)) = γ(Γ(R)).

DOMINATION NUMBER IN THE ANNIHILATING-IDEAL GRAPHS 229
Proof. The result follows from Part (2) of Proposition 2.1 and Theorems 2.4
and 2.5.
In the following theorem the domination number of bipartite annihilating-ideal
graphs is given.
Theorem 2.6. If AG(R) is a bipartite graph, then γ(AG(R)) 6 2.
Proof. If AG(R) is bipartite, it follows from [1, Theorem 27] that one of the
following cases occurs: (a): AG(R) is a sta r graph; (b): AG(R) is the path of
order 4; (c): Nil(R) = Soc(R). If (a) or (b) ha ppen, then we are done. Suppose
that Nil(R) = Soc(R). If R is reduced, then the result follows from Corollar y 2.1.
If R is nonreduced, then [1, Theorem 20] completes the proof.
The next theorem is on the domination number of the annihilating-ideal graphs
of Artinian rings.
Theorem 2.7. Let R be an Artinian ring and R F
1
× F
2
, where F
1
and F
2
are two ﬁelds. Then γ(AG(R)) = γ
t
(AG(R)) = | Min(R)|.
Proof. Since R is Artinian, we deduce that each ideal of R is an annihilating-
ideal. So, the set of maximal elements of A(R) and Max(R) are eq ual. By [4,
Theorem 8 .7], R
=
R
1
× · · · × R
k
, where (R
i
, m
i
) is an Artinian local ring, for
i = 1, . . . , k. Let Max(R) = {n
i
= R
1
× · · · × R
i1
× m
i
× R
i+1
× · · · × R
k
| 1 6
i 6 k}. By Theorem 2.2, γ
t
(AG(R)) = | Max(R)|. In the sequel, we prove that
γ(AG(R)) = k. Assume to the contrary, the se t {J
1
, . . . , J
k1
} is a dominating
set for AG(R). Since R F
1
× F
2
, where F
1
and F
2
are two ﬁelds, we ﬁnd that
J
i
n
s
= J
i
n
t
= 0, for some i, t, s, where 1 6 i 6 k 1 and 1 6 t, s 6 k. This means
that J
i
The c ondition of R to be an Artinian ring in the previous theorem is necessary;
see the next example.
Example 2 .1. Let R =
k[x,y,z]
(xy,xz,yz)
, where k is a ﬁeld and x, y and z are inde-
terminates. Then γ(AG (R)) = 2 but | Min(R)| > 2.
Theorem 2.7 gives the following immediate corollary.
Corollary 2.3. Let R be an Artinian ring and R F
1
× F
2
, where F
1
and
F
2
are two ﬁelds. Then γ(AG(R)) = γ(Γ(R)) = | Min(R)|.
Proof. The result follows from Theorem 2.7 and [10, Theor e m 11].
Example 2.2. Let n be a natural numbe r and n = p
n
1
1
p
n
2
2
. . . p
n
m
m
, where p
i
’s
are distinct primes and n
i
’s are natur al numbers. Then one of the following holds:
(i) γ(AG(Z
n
)) = 1 if and only if either n = p
1
p
2
or n = p
n
1
1
, where n
1
> 1.
(ii) γ(AG(Z
n
)) = 2 if and only if m = 2 and either n
1
> 1 or n
2
> 1.
(iii) If m > 3, then γ(AG(Z
n
)) = m.
The following theorem provides an upper b ound for the domination number of
the annihilating-ideal graph of a Noetherian ring.

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### Cites background from "Domination number in the annihilati..."

• ...Also, we note a mistake of [28] and we characterize the domination of a ring in which the zero ideal is a fixed-place ideal and domination of AG(X) in which X is almost discrete and finally we prove that dt(AG(R)) is finite, if and only if dtt(AG(R)) is finite; if and only if Min(R) is finite....

[...]

• ...Also, First the annihilatingideal graph has been introduced and studied in [19] and then it has been studied in several articles; see [20, 9, 2, 1, 27, 22, 28]....

[...]

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###### Q1. What contributions have the authors mentioned in the paper "Domination number in the annihilating-ideal graphs of commutative rings" ?

In this paper, the authors study the domination number of AG ( R ) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

Since R ≇ F1 × F2, where F1 and F2 are two fields, the authors find that Jins = Jint = 0, for some i, t, s, where 1 6 i 6 k − 1 and 1 6 t, s 6 k.

The zero-divisor graph of R, Γ(R), is a graph with the vertex set Z(R)r {0} and two distinct vertices x and y are adjacent if and only if xy = 0.

A prime ideal p is said to be an associated prime ideal of a commutative Noetherian ring R, if there exists a nonzero element x in R such that p = ann(x).

By [6, Theorem 2.2], there is a vertex of AG(R) which is adjacent to every other vertex if and only if either R ∼= F × D, where F is a field and D is an integral domain, or Z(R) is an annihilator ideal.

by [3, Theorem 2.5], there is a vertex of Γ(R) which is adjacent to every other vertex if and only if either R ∼= Z2 × D, where D is an integral domain, or Z(R) is an annihilator ideal.

The authors may assume that γ(AG(R1)) = m and γ(AG(R2)) = n, for some positive integers m and n. Let {I1, . . . , Im} and {J1, . . . , Jn} be two minimal dominating sets in AG(R1) and AG(R1), respectively.

For a ring R, which is a product of two (nonzero) rings, one of the following holds:(i) If R ∼= F × D, where F is a field and D is an integral domain, then γ(AG(R)) = 1.(ii) If R ∼= D1 × D2, where D1 and D2 are integral domains which are not fields, then γ(AG(R)) = 2.(iii)

Since n > 3, the ideals pi and pj , for i 6= j are not adjacent (from pipj = 0 ⊆ pk it would follow that pi ⊆ pk, or pj ⊆ pk which is not true).

For reduced rings with finitely many minimal primes and Artinian rings, the domination number of the annihilating-ideal graphs is given.

With no loss of generality, one can assume that γ(AG(R1)) < ∞. Suppose that γ(AG(R1)) = n and {I1, . . . , In} is a minimal dominating set of AG(R1).

The study of algebraic structures, using the properties of graphs, have become an exciting research topic in the past twenty years, leading to many interesting results and questions.

By Ass(R) the authors denote the set of all associated prime ideals of R. A ring R is said to be reduced, if it has no nonzero nilpotent element or equivalently ⋂