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On the co-annihilating ideal graph of commutative rings
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In this research article, we consider a commutative ring with unity denoted as
S
\mathcal {S}
and
A
(
S
)
A(\mathcal {S})
as the collection of all annihilating ideals of
S
\mathcal {S}
. We introduce a graph called the co-annihilating ideal graph of
S
\mathcal {S}
, denoted as
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
, with vertices taken from the set
A
(
S
)
∗
A(\mathcal {S})^*
(including all non-zero annihilating ideals of
S
\mathcal {S}
) and in this graph two distinct vertices
X
\mathcal {X}
and
Y
\mathcal {Y}
are adjacent if and only if
X
⊈
X
Y
\mathcal {X} \nsubseteq \mathcal {XY}
and
Y
⊈
X
Y
\mathcal {Y} \nsubseteq \mathcal {XY}
. We establish the result that
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
is connected with a diameter of at most two and a girth of at most four, if
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
contains a cycle. Additionally, we identify rings in which the co-annihilating ideal graph takes the form of either a complete graph or a star graph. Furthermore, we undertake the task of classifying all Artinian rings
S
\mathcal {S}
where the co-annihilating ideal graph
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
can be matched with well-known existing graphs.
American Mathematical Society
Title: On the co-annihilating ideal graph of commutative rings
Description:
In this research article, we consider a commutative ring with unity denoted as
S
\mathcal {S}
and
A
(
S
)
A(\mathcal {S})
as the collection of all annihilating ideals of
S
\mathcal {S}
.
We introduce a graph called the co-annihilating ideal graph of
S
\mathcal {S}
, denoted as
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
, with vertices taken from the set
A
(
S
)
∗
A(\mathcal {S})^*
(including all non-zero annihilating ideals of
S
\mathcal {S}
) and in this graph two distinct vertices
X
\mathcal {X}
and
Y
\mathcal {Y}
are adjacent if and only if
X
⊈
X
Y
\mathcal {X} \nsubseteq \mathcal {XY}
and
Y
⊈
X
Y
\mathcal {Y} \nsubseteq \mathcal {XY}
.
We establish the result that
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
is connected with a diameter of at most two and a girth of at most four, if
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
contains a cycle.
Additionally, we identify rings in which the co-annihilating ideal graph takes the form of either a complete graph or a star graph.
Furthermore, we undertake the task of classifying all Artinian rings
S
\mathcal {S}
where the co-annihilating ideal graph
C
A
G
(
S
)
\mathbb {CAG}(\mathcal {S})
can be matched with well-known existing graphs.
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