|
Abstract
|
Let R be a commutative ring with identity. An ideal I of a ring
R is called an annihilating ideal if there exists r 2 R n f0g such that Ir = (0)
and an ideal I of R is called an essential ideal if I has non-zero intersection
with every other non-zero ideal of R. The sum-annihilating essential ideal
graph of R, denoted by AER, is a graph whose vertex set is the set of all
non-zero annihilating ideals and two vertices I and J are adjacent whenever
Ann(I) + Ann(J) is an essential ideal. In this paper we initiate the study of
the sum-annihilating essential ideal graph. We rst characterize all rings whose
sum-annihilating essential ideal graphs are stars or complete graphs and then we
establish sharp bounds on the domination number of this graph. Furthermore,
we determine all isomorphism classes of Artinian rings whose sum-annihilating
essential ideal graphs have genus zero or one.
|