International Journal of Algebra, Vol. 4, 2010, no. 5, 209 - 220
h-Prime and h-Semiprime Ideals in
Semirings and Γ-Semirings
Sujit Kumar Sardar
Department of Mathematics, Jadavpur University
Kolkata-700032, West Bengal,India
sksardarjumath@gmail.com
Bibhas Chandra Saha
Chandas Mahavidyalaya
Khujutipara, Birbhum
West Bengal,India, PIN 731215
bibhas
−
sh@yahoo.co.in
K. P. Shum
Department of Mathematics
The University of Hong Kong
Pokfulam Road, Hong Kong, China (SAR)
kpshum@maths.hku.hk
Abstract
The concepts of h-prime ideals and h-semiprime ideals in semirings
and Γ-semirings are introduced so that their properties are studied. In
particular, the relationships between Γ-semirings and its related opera-
tor semirings are described in terms of the h-closure; the h-prime and
h-semiprime ideals. These results will be used to obtain some other new
results such as the inclusion preserving bijections between the h-prime
(or h-semiprime) ideals of a Γ-semiring and its related operator semir-
ings. Moreover, the h-regularity and the H-Noetherian Γ-semirings will
be characterized. Some recent results given by T. K. Dutta and S. K.
Sardar in semiprime ideals and irreducible ideals of a Γ- Semirings and
extended and generlized.
Mathematics Subject Classification: 16Y60, 16Y99, 20N10
Keywords: h-prime ideals; h-semiprime ideals; Γ-semiringss; Operator
semirings; H-Noetherian Γ-semirings ; h-regularity
210 S. K. Sardar, B. Ch. Saha and K. P. Shum
1 Introduction
It is well known that in the theory of Γ-semirings , the properties of their
ideals, prime ideals and semiprime ideals play an important role, however the
properties of an ideal in a semiring are sometimes differ from the properties of
a ring ideals . In order to amend this gap, the concepts of k-ideals [5] and h-
ideals in a semiring were first considered by D. R. LaTorre in 1965 [6]. For the
definition of h-ideals in Γ-semirings, the reader is referred to the recent paper
of T. K. Dutta and S. K. Sardar in [2]. The notions of operator semirings and
Γ-semirings were also introduced by them. Moreover, the properties of prime
ideals [1] and semiprime ideals in Γ-semirings were studied and discussed by
them in [3].
By using the concept of k-ideal [5], D. M. Olson et al have considered the
pre-prime and pre-semiprime ideal of a semiring in [9, 10]. In this aspect, S.
K.Sardar and B. C. Saha developed the same notion in Γ-semirings in [11] . In
this paper, by using the concept of h-ideals, we also introduce the notions of
prime h-ideals and investigate the properties of h-prime ideals and h-semiprime
ideals in semirings and Γ-semirings. We therefore derive a new operation by
taking the h-closure of a given operation which is commutating with each of
the following operations “∗
, ∗,+
, + ” given in [2]. Finally, the so called
H-Noetherian Γ-semirings will be characterized. Some related results obtained
by T. K. Dutta and S. K. Sardar in operator semirings of a Γ-semiring [2,4]
will be hence enriched and amplified.
For some basic results and definitions which were not given in this paper,
the reader is referred to the monograph of J. S. Golan in semirings, see [5].
2 h-prime and h-semiprime ideals in semirings
Definition 2.1 [2, 6] An ideal I of a semiring S (respectively Γ-semring)
is called an h-ideal of S if x + y
1
+ z = y
2
+ z; y
1
,y
2
∈I; x, z ∈S implies x ∈I.
Definition 2.2 [12] The h-closure
A of A is defined by A = {x ∈ S :
x + a
1
+ z = a
2
+ z for some a
1
,a
2
∈ A, z ∈ S}
For the sake of convenience, we denote
A by H(A).
Definition 2.3 An ideal P of a semiring S is said to be an h-prime ideal
of S if IJ⊆H(P) implies I⊆H(P) or J⊆H(P), where I, J are ideals of S.
Similarly, we have the following definition.
Definition 2.4 An ideal P of a semiring S is said to be an h-semiprime
ideal of S if I
2
⊆H(P) implies I ⊆H(P), where I is an ideal of S.
h-prime and h-semiprime ideals in semirings 211
The following results follow immediately from the above definitions .
Proposition 2.5 .The following statements hold in a semiring S.
• (i) An ideal P of S is h-prime if and only if H(P ) is prime in S.
• (ii) An ideal P of S is h-semiprime if and only if H(P) is semiprime in
S.
Note. It can be easily seen that an h-prime ideal P of a semiring S is
h-semiprime but this statement does not hold conversely.
By Proposition 7.4 in [5] and by Proposition 2.5, we can easily deduce the
following characterization theorem for an h-prime ideal in a semiring.
Theorem 2.6 Let S be a semiring with identity and I an ideal of S. Then
the following statements are equivalent:
i) I is h-prime,
ii) for any a, b ∈S, aSb ⊆H(I) implies that a ∈H(I) or b ∈H(I),
iii) for any a, b ∈S, <a>.<b>⊆H(I) implies that a ∈H(I) or b ∈H(I).
Analogously, one can prove the following characterization theorem for h-
semiprime ideals in semirings.
Theorem 2.7 Let S be a semiring with unity. Then the following state-
ments are equivalent for any ideal I of S,
i) I is h-semiprime.
ii) For a∈S such that aSa ⊆H(I) implies a∈H(I),
iii) <a>
2
⊆H(I) implies that a ∈H(I).
3 h-prime and h-semiprime ideals in Γ-semirings
Throughout this section, unless otherwise stated, S is a Γ-semiring . We also
use L and R to denote the left and right operator semirings, respectively. Now,
we define H(I):={x ∈S: x + y
1
+ z = y
2
+ z, for some y
1
,y
2
∈I, z ∈S } and
we want to show that H(I) is an h-ideal of S with I ⊆ H(I).
For the sake of brevity, we just denote the h-closure of I in a Γ-semiring
S by H(I). Thus if A and B are ideals of the Γ-semiring S with A⊆B, then
212 S. K. Sardar, B. Ch. Saha and K. P. Shum
H(A)⊆H(B).
We now give here the definitions of h-prime ideals and h-semiprime ideals
in a Γ-semiring s. it is clear that these concepts are more general than the
usual concepts of prime and semiprime ideals.
Definition 3.1 An ideal P of a Γ-semiring S is called an h-prime ideal of
S if IΓJ⊆H(P) implies I⊆H(P) or J⊆H(P), where I and J are ideals of S.
Definition 3.2 An ideal P of a Γ-semiring S is said to be an h-semiprime
ideal of S if for any ideal I of S, IΓI⊆H(P) implies I⊆H(P).
The concepts of h-prime ideal and h-semiprime ideals of a semiring or a
Γ-semiring S are proper generalizations of the concepts of prime ideals and
semiprime ideals of S. Thus, an h-prime ( h-semiprime) ideal of a semiring or
a Γ-semiringS is not necessarily a prime(semiprime) ideal of S and vice-versa.
This observation can be easily seen in the following examples:
Example 3.3 Let S=Z
+
0
be the set of all nonnegative integers. Then S is
a semiring with respect to the usual addition and multiplication. Let P=2Z
+
0
\
{2, 4, 6}. Then P is an ideal of S but not a prime ideal of the semiring S.
Now H(P)=2Z
+
0
is clearly a prime ideal of S. This example illustrates that
there exists an h-prime ideal of S which is not a prime ideal of S.
Example 3.4 In the above example, If we let P=Z
+
0
\{1}, then P is a
prime ideal of the semiring S. Now it is clear that H(P)=Z
+
0
is a not a prime
ideal of S because according to the definition a prime ideal of a semiring s,
H(P ) is a proper ideal [5]) of S, consequently, P is not an h-prime ideal of S.
Example 3.5 Let S=Z
−
0
be the set of all non-positive integers and Γ=
{a, b}. Then with the addition defined as follows:
+
ab
a ab
b
ba
Now, one can easily see that Γ is a commutative semigroup. If we define the
ternary composition
S × Γ × S → S as follows:
(x, a, y) → 0 and
(x, b, y) →−(xy)
h-prime and h-semiprime ideals in semirings 213
then S forms a Γ-semiring. Now let P=2Z
−
0
\{-2}. Then P is an ideal of S
but not a prime ideal of the Γ-semiring S. But H(P)=2Z
+
0
is a prime ideal of
S. Hence P is an h-prime ideal of S.
We now state below some known properties of h-prime ideals and h-semiprime
ideals of a Γ-semiring.
Proposition 3.6 An ideal P of a Γ-semiring S is h-prime if and only if H(P)
is prime in S.
Proposition 3.7 Any ideal P of Γ-semiring S is h-semiprime if and only if
H(P) is semiprime in S.
Note. It is well known that an h-prime ideal I of a Γ-semiring S is h-
semiprime but the converse does not hold.
Before we proceed to establish the main results of this paper, we first
introduce the concept of the left operator semirings (respectively right operator
semiring) of a Γ-semiring which have been described by Dutta and Sardar in
[4].
Definition 3.8 Let S be a Γ-semiring and F a free additive commutative
semigroup generated by S×Γ . Define the following relation on F :
(
m
j=1
(x
i
; α
i
),
n
j=1
(y
j
; β
j
)) ∈ ρ
if and only if
m
j=1
x
i
α
i
a =
n
j=1
y
j
β
j
a
∀a ∈ S.
Then the relation ρ can be easily verified to be a congruence on F and so
F /ρ forms a semiring under the following multiplication
(
m
j=1
[x
i
; α
i
]) • (
n
j=1
([y
j
; β
j
]) =
i,j
[x
i
α
i
y
j
; β
j
],
where
m
j=1
[x
i
; α
i
],
n
j=1
[y
j
; β
j
] are the classes of ρ which contains
m
j=1
(x
i
; α
i
)
and
n
j=1
(y
j
; β
j
), respectively. We denote this semiring by L which is called the