Boll. Unione Mat. Ital.
DOI 10.1007/s40574-017-0148-7
An intersection condition for graded prime ideals
Khaldoun Al-Zoubi
1
· Feda’a Qarqaz
1
Received: 8 April 2017 / Accepted: 20 October 2017
© Unione Matematica Italiana 2017
Abstract Let G be a group with identity e and let R be a commutative G-graded ring. In
this paper, we will investigate commutative graded rings which satisfy the condition (∗). We
say that a graded ring R satisfy the condition (∗) if P is a graded prime ideal of R and if
{ I
α
}
α∈
is a family of graded ideals of R, then P contains ∩
α∈
I
α
only if P contains some
I
α
.
Keywords Graded prime ideals · Intersection condition · Graded ideals
Mathematics Subject Classification 13A02 · 16W50
1 Introduction and preliminaries
Throughout this paper, all rings are assumed to be commutative with identity elements. The
concept of graded prime ideal was introduced and studied in [1, 2, 6, 8] as a generalization of
the notion of prime ideal. Let R be a G-graded ring. A proper graded ideal P of R is said
to be a graded prime ideal (or gr-prime ideal) of R if whenever r and s are homogeneous
elements of R such that rs ∈ P , then either r ∈ P or s ∈ P . In [6, Proposition 1.4], the
authors proved the following property : If R is a G-graded ring, I
1
, I
2
,..., I
n
a finite number
of graded ideals of R, and P a graded prime ideal of R such that ∩
n
i =1
I
i
⊆ P, then I
j
⊆ P
for some j ∈{1, 2,..., n}. In this paper, we continue our research on intersection conditions
for graded prime ideals. We study a graded ring R with property: (∗) If P is a graded prime
ideal of R and if { I
α
}
α∈
is a family of graded ideals of R, then P contains ∩
α∈
I
α
only if
P contains some I
α
.
First, we recall some basic properties of graded rings and modules which will be used in
the sequel. We refer to [4–6] for these basic properties and more information on graded rings
B Khaldoun Al-Zoubi
kfzoubi@just.edu.jo
1
Department of Mathematics and Statistics, Jordan University of Science and Technology,
P.O.Box 3030, Irbid 22110, Jordan
123