The triple zero graph of a commutative ring
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Date
2021
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ankara Üniversitesi
Abstract
Let
R
be a commutative ring with non-zero identity. We define the set of triple zero elements of
R
by
T
Z
(
R
)
=
{
a
∈
Z
(
R
)
∗
:
there exists
b
,
c
∈
R
∖
{
0
}
such that
a
b
c
=
0
,
a
b
≠
0
,
a
c
≠
0
,
b
c
≠
0
}
.
In this paper, we introduce and study some properties of the triple zero graph of
R
which is an undirected graph
T
Z
Γ
(
R
)
with vertices
T
Z
(
R
)
,
and two vertices
a
and
b
are adjacent if and only if
a
b
≠
0
and there exists a non-zero element
c
of
R
such that
a
c
≠
0
,
b
c
≠
0
, and
a
b
c
=
0
. We investigate some properties of the triple zero graph of a general ZPI-ring
R
,
we prove that
d
i
a
m
(
T
Z
Γ
(
R
)
)
∈
{
0
,
1
,
2
}
and
g
r
(
G
)
∈
{
3
,
∞
}
.
Description
Keywords
Triple zero graph, Zero-divisor graph, 2-absorbing ideal