Title of article :
THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING
Author/Authors :
Alibemani, Abolfazl Faculty of Mathematical Sciences - Shahrood University of Technology , Hashemi, Ebrahim Faculty of Mathematical Sciences - Shahrood University of Technology , Alhevaz, Abdollah Faculty of Mathematical Sciences - Shahrood University of Technology
Abstract :
Let
R
R be an associative ring with
1
≠
0
1≠0 which is not a domain. Let
A
(
R
)
∗
=
{
I
⊆
R
∣
I
i
s
a
l
e
f
t
o
r
r
i
g
h
t
i
d
e
a
l
o
f
R
a
n
d
l
.
a
n
n
(
I
)
∪
r
.
a
n
n
(
I
)
≠
0
}
∖
{
0
}
A(R)
∗
={I⊆R ∣ I is a left or right ideal of R and l.ann(I)∪r.ann(I)≠0}∖{0}. The total graph of annihilating one-sided ideals of
R
R, denoted by
Ω
(
R
)
Ω(R), is a graph with the vertex set
A
(
R
)
∗
A(R)
∗
and two distinct vertices
I
I and
J
J are adjacent if
l
.
a
n
n
(
I
+
J
)
∪
r
.
a
n
n
(
I
+
J
)
≠
0
l.ann(I+J)∪r.ann(I+J)≠0. In this paper, we study the relations between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose graphs are disconnected. Also, we study diameter, girth, independence number, domination number and planarity of this graph
Keywords :
Total graph , diameter , reversible ring , semicommutative ring , skew polynomial ring
Journal title :
International Electronic Journal of Algebra
