Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
On the co-annihilating ideal graph of commutative rings
View through CrossRef
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.
Related Results
Bipartite through Prescribed Median and Antimedian of a Commutative Ring with Respect to an Ideal
Bipartite through Prescribed Median and Antimedian of a Commutative Ring with Respect to an Ideal
Introduction: There are plenty of ways of partners with arithmetical constructions. Some of them to make reference to are bipartite from gatherings, median and anti – median from c...
Effects of cleaning in Saturn's rings
Effects of cleaning in Saturn's rings
Saturn's rings are well known for many good reasons, one of them being their brightness. Made of almost 99% water ice, they are by far the most ice-rich object of the solar system,...
Graph convolutional neural networks for 3D data analysis
Graph convolutional neural networks for 3D data analysis
(English) Deep Learning allows the extraction of complex features directly from raw input data, eliminating the need for hand-crafted features from the classical Machine Learning p...
Bilangan Terhubung Titik Pelangi pada Graf Garis dan Graf Tengah dari Hasil Operasi Comb Graf Bintang C<sub>3</sub> dan Graf Bintang S<sub>n</sub>
Bilangan Terhubung Titik Pelangi pada Graf Garis dan Graf Tengah dari Hasil Operasi Comb Graf Bintang C<sub>3</sub> dan Graf Bintang S<sub>n</sub>
Penelitian ini bertujuan menentukan bilangan terhubung titik pelangi (rainbow vertex connection number) pada graf garis dan graf tengah yang diperoleh dari hasil operasi comb antar...
On the Commutative Operation of Approximate CMOS Ripple Carry Adders (RCAs)
On the Commutative Operation of Approximate CMOS Ripple Carry Adders (RCAs)
<p>Approximate cells can be used to design Ripple Carry Adders (RCAs) for realizing approximate addition in energy-efficient CMOS digital circuits. As inputs of approximate c...
On the Commutative Operation of Approximate CMOS Ripple Carry Adders (RCAs)
On the Commutative Operation of Approximate CMOS Ripple Carry Adders (RCAs)
Approximate cells can be used to design Ripple Carry Adders (RCAs) for
realizing approximate addition in energy-efficient CMOS digital
circuits. As inputs of approximate cells coul...
Determinants of some special matrices over commutative finite chain rings
Determinants of some special matrices over commutative finite chain rings
AbstractCirculant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many ca...
On the dominating sets of the complement of the annihilating ideal graph of a commutative ring
On the dominating sets of the complement of the annihilating ideal graph of a commutative ring
Let R be a commutative ring with identity which is not an integral domain. Let A(R)∗ denote the collection of all nonzero annihilating ideals of R and AG(R) denote the annihilating...