EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 2, 2018, 449-456
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Almost prime ideal in gamma near ring

Waheed Ahmad Khan1, Adnan Muhammad1, Abdelghani Taouti2,∗,
Jameel Maki3

1 University of Education Lahore, Attock Campus, Pakistan
2 ETS-Maths and NS Engineering Division, HCT, University City P.O. Box 7947, Sharjah,
United Arab Emirates
3 Department of Mathematics, R.I.T Dubai. P. O. Box 341055, Dubai, United Arab Emirates

Abstract. In this manuscript we introduce the notion of almost prime ideals in Γ-near-rings along
with few of their characterizations. We also present the interesting relations among almost prime,
prime and primary ideal in Γ-nearrings.

2010 Mathematics Subject Classifications: 13A05, 13A18, 12J20

Key Words and Phrases: Γ-near-rings, prime ideals, almost prime ideals.

1. Introduction and Preliminaries

Recently, the generalization of prime ideal i.e., almost prime ideal in commutative rings
has been introduced and discussed by Srikant M. Bhatwadekar and Pramod K. Sharma
( See [3]). Following [3], an ideal I of a ring R is said to be an almost prime if for all
a, b ∈ R implies ab ∈ I − I2 either a ∈ I or b ∈ I. All prime and idempotent ideals are
almost prime [3]. It has been proved that every almost prime ideal in a noetherian domain
R is primary [3]. Further to this, almost primary ideals in rings have been introduced by
A. K. Jabbar and C. A. Ahmed in [12], a proper ideal A of a ring R is an almost primary
ideal if for a, b ∈ R such that ab ∈ A−A2, then a ∈ A or b ∈ A, for some positive integer
n [12]. In [12], authors have also discussed several characterizations of almost primary
ideals. It is evident that primary ideals, almost prime ideals and idempotent ideals of a
ring R are almost primary ideals, but the converse is not true in each case. Notion of
weakly prime element (author called it a prime) was introduced by Steven Galovich while
studying the property of unique factorization of rings with zero divisors [10]. Following
[10], let r 6= 0 be in R than r is prime if, whenever r divides ab where ab 6= 0, then r
divides a or r divides b. Author established the fundamental results: (i) In [10], author also

∗Corresponding author.

Email addresses: sirwak2003@yahoo.com (W. A. Khan),
adnanmuhammad216@gmail.com (A. Muhammad), ganitaouti@yahoo.com.au (A. Taouti),
jamcad@rit.edu (J. Maki)

http://www.ejpam.com 449 c© 2018 EJPAM All rights reserved.



A. Taouti et al. / Eur. J. Pure Appl. Math, 11 (2) (2018), 449-456 450

showed that every irreducible is a prime, (ii) every irreducible in R is a zero divisor [10],
(iii) every irreducible element of R is nilpotent, and (iv) every nonunit in R is nilpotent.
Consequently the author declared the unique maximal ideal consists of nonunit elements
[10]. In [1], authors declare that (which was named prime by Galovich in [10]) a nonzero
nonunit p ∈ R is weakly prime if p|ab 6= 0 implies p|a or p|b. Consequently, an ideal I of
a commutative ring R is called a weakly prime if 0 6= ab ∈ I implies a ∈ I or b ∈ I, and
also p is weakly prime iff (p) is weakly prime [1]. Following [2], P is weakly prime ideal if
and only if 0 6= AB ⊆ P , A and B ideals of R, implies A ⊆ P or B ⊆ P . Further to this,
every weakly prime ideal is an almost prime ideal.
We call an algebraic system N with two binary operation ” + ” and ”.” (right) near-ring
if it is a group (not necessarily abelian) under addition, and N is associative group under
multiplication and distribution of multiplication over addition on the right holds i.e., for
any x, y, z ∈ N , it satisfies that (x + y)z = (xz) + (yz)[15]. Likewise, a left near-ring
can be defined by replacing the right distributive law by the equivalent left distributive
law. Suppose N is a left near-ring with binary operation ” + ” and ”.” then a subset I
is said to be an ideal if (i) (I,+) is a normal subgroup of a (N,+), (ii) For each n ∈ N ,
i ∈ I, ni ∈ I i.e., NI ⊆ I, and (iii) (n1 + i)n2 n1n2 ∈ I for each n1, n2 ∈ N and i ∈ I.
But A. Frohlich [9] showed that for d.g. near-rings the third condition is equivalent to
in ∈ I i.e., IN ⊆ I. Hence a subset I is a right (left) ideal if I satisfies the first and third
(second) conditions. A proper ideal P of a near ring N is prime if for ideals A and B of
N , AB ⊆ P implies A ⊆ P or B ⊆ P . An ideal P of a near-ring N is a completely prime
(prime ideal of type-2) if for all x, y ∈ N , xy ∈ P implies x ∈ P or y ∈ P . Almost prime
ideals in near rings have been endorsed by B. Elavarasan (see [8]). A proper ideal P of a
near ring N is said to be almost prime if for any ideals A and B of N such that AB ⊆ P
and AB * P 2, we have A ⊆ P or B ⊆ P [8]. The author established few relationships
between almost prime and prime ideals [8]. Weakly prime ideals in near rings have been
introduced by P. Dheena and B. Elavarasan [6], a proper ideal P of near ring N is said
to be weakly prime if 0 6= AB ⊆ P , A and B are ideals of N , implies A ⊆ P or B ⊆ P .
Clearly, every prime ideal is weakly prime and {0} is always weakly prime ideal of a near
ring N . Also every prime ideal is a weakly prime, and a weakly prime ideal is an almost
prime ideal. An ideal I of a near ring N is said to be a completely prime ideal if x, y ∈ N ,
xy ∈ I implies x ∈ I or y ∈ I [11]. Similarly, an ideal of a near ring N is said to be
primary ideal of N if x, y ∈ N , xy ∈ I implies x ∈ I or ym ∈ I for some m ∈ Z. An
ideal I of a near ring N is called a completely semiprime ideal of a near ring N if y2 ∈ I
implies y ∈ I for all y ∈ N [11]. Further to this, almost prime ideals in near rings have
been endorsed by B. Elavarasan (see [8]). A proper ideal P of a near ring N is said to be
almost prime if for any ideals A and B of N such that AB ⊆ P and AB * P 2, we have
A ⊆ P or B ⊆ P [8]. The author established few relationships between almost prime and
prime ideals [8]. Number of ideals in near ring have been introduced and discussed such
as completely prime, primary, completely primary and so on. Following [11], an ideal I of
a near ring N is said to be a completely prime ideal if x, y ∈ N , xy ∈ I implies x ∈ I or
y ∈ I [11]. Similarly, an ideal of a near ring N is said to be primary ideal of N if x, y ∈ N ,
xy ∈ I implies x ∈ I or ym ∈ I for some m ∈ Z. An ideal I of a near ring N is called a



A. Taouti et al. / Eur. J. Pure Appl. Math, 11 (2) (2018), 449-456 451

completely semiprime ideal of a near ring N if y2 ∈ I implies y ∈ I for all y ∈ N [11].
The ideal theory is the most important part of algebra, different types of ideals in rings
have been discussed in the literature. A right (left) ideal of a Γ-ring M is an additive
subgroup I of M such that IΓM ⊆ I (MΓI ⊆ I). If I is both a right and a left ideal, then
we say that I is an ideal or a two-sided ideal of M . In rings, an ideal P is prime ideal if
and only if A and B are ideals in M such that AB ⊆ P , then A ⊆ P or B ⊆ P [13]. The
prime ideals of the Γn,m -ring Mm,n are the sets Pm,n corresponding to the prime ideals
P of the Γ-ring M [13]. If P is an ideal in a Γ-ring M then, (i) Ideal P is a prime ideal
of M , (ii) If a, b ∈M and aΓMΓb ⊆ P then either a ∈ P or b ∈ P , (iii) If ideal generated
by < a > and < b > are called principal ideals in M and < a > Γ < b >⊆ P , then a ∈ P
or b ∈ P , (iv) If U and V are right ideals in M with UΓV ⊆ P , then U ⊆ P or V ⊆ P ,
(v) If U and V are left ideals in M with UΓV ⊆ P , either U ⊆ P or V ⊆ P [16].
Γ-near rings were introduced by Satyanarayana Bhavanari (see [14], [15]). A subset A of
a Γ-near-ring M is called a left (resp. right) ideal of M if (A,+) is a normal divisor of
(M,+), uα(x + v) − uαv ∈ A (resp. xαu ∈ A ) for all x ∈ A, α ∈ Γ and u, v ∈ M .
An ideal P of Γ-near ring (M,+, (.)Γ) is called prime, if for every two ideals I, J of M ,
IΓJ ⊆ P implies I ⊆ P or J ⊆ P . An ideal P of a Γ-near-ring N is called a completely
primary ideal if for a, b ∈ N and γ ∈ Γ such that aγb ∈ P implies that a ∈ P or b ∈ P , for
some positive integer n [17]. If an ideal I of Γ-near-ring M is maximal, then it is prime or
MΓM = I [7]. If (M,+, (.)Γ) is a Γ-near-ring such that for any γ ∈ Γ there is an element
which is Γ-unit, then every maximal ideal I of M is prime [7]. For every ideal I of Γ-near-
ring M exists prime minimal ideal of I [7]. In this note first we introduce the notion of
almost prime ideals in Γ-near-rings along with few of their characterizations. Finally, we
present the interesting relations of an almost prime with the prime and primary ideal in
Γ-near-rings.

2. Almost prime ideal in Γ-near-ring

In this section we introduce almost prime ideal in Γ-near-rings. Furthermore, we also
present its implications with the some ideals, we start with the following definition.
Definition 1. Let M be Γ-near-ring and P be a prime ideal of M then P is almost prime
ideal if a, b ∈ R, ab ∈ P − PΓP , either a ∈ P or b ∈ P .
Example 1. Suppose Z8 = {0, 1, 2, 3, 4, 5, 6, 7} and Γ = {0, 2, 4}. Let P = 2Z8 = {0, 2, 4}
be a prime ideal in Z8 and consider PΓP = {0, 6}, P −PΓP = {2, 4}. Here 2, 3 ∈ Z8 and
2.2.3 = 4 ∈ P − PΓP where 2 ∈ P and 3 6∈ P . Similarly we can check for other elements
as well. Hence P is an almost prime ideal in Γ-near ring.
Example 2. Suppose R is a Γ-near ring of algebraic integers such that the integral closure
of Z in C. Suppose that I be a radical ideal of R say IΓI = I, if α ∈ I then β ∈ R exist
such that βΓβ = α. Since βΓβ = α ∈ I, β ∈ I implies I = IΓI.
Example 3. Consider the near ring N = {0, 1, 2, 3} and Γ = {0, 2} such that addition
and multiplication defined as follow.



A. Taouti et al. / Eur. J. Pure Appl. Math, 11 (2) (2018), 449-456 452
+ 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0



· 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1


Suppose P = {0, 2} = 2N be a prime ideal of N because for all a, b ∈ N and aγb ∈ P

implies a ∈ P or b ∈ P . As PΓP = {0} then P − PΓP = {2}, then for all a, b ∈ N such
that aγb ∈ P − PΓP either a ∈ P or b ∈ P which is almost prime ideal.
Preposition 1. Every prime ideal in a Γ-near ring is almost prime ideal. Proof. Suppose
P be a prime ideal of Γ-near ring but not an almost prime. Assume aγb ∈ P − PΓP ,
implies aγb ∈ P . If aγb 6∈ PΓP implies a ∈ P or b ∈ P then contradiction arise to our
supposition. Hence P must be a prime.
Remark 1. If I is a maximal ideal of Γ-near-ring M then it is prime or MΓM = I.
Supporting the above remark 1, we present the below example.
Example 4. Let M = {0, 1, 2, 3} is a Γ-near-ring where Γ = {0, 2} and ideal I = 2M =
{0, 2} that is maximal in M . Obviously I is prime ideal in M also MΓM = I.
Lemma 1. Suppose N is a Γ-near-ring and for any γ ∈ Γ there is an element which is
Γ-unit then every maximal ideal I of M is prime.
Proof. If for one γ ∈ Γ the element e is γ-one of M then MγM = {m1γm2 : m1;m2 ∈
M} = M since for any m ∈ M , m = mγe. Because M 6= I the equation is not true
MΓM = I. When M = I or M = 0 then equation is true so M is simple and MΓM 6= 0,
as a result M is prime.
Preposition 2. Suppose I be a P -primary ideal of a Γ-near ring such that PΓP = IΓI
implies I is an almost prime.
Proof. Suppose a, b ∈ R, aγb ∈ I − IΓI, a 6∈ I and b 6∈ I. As a 6∈ I and I is a P -primary
ideal it implies that b ∈ P . Also a ∈ P thus aγb ∈ PΓP = IΓI, which is a contradiction.
Lemma 2. Suppose that R be a near integral domain and c be a nonzero nonunit element
of R. If element c is other than prime element then there exist a 6∈ RΓc, b 6∈ RΓc such
that aγb ∈ RΓc but aγb 6∈ RΓc2.
Proof. Suppose an ideal Rc is not prime then there exist a 6∈ RΓc, b 6∈ RΓc such that
aγb ∈ RΓc. If the case aγb ∈ RΓc2 then for d = (b + c)γ 6∈ RΓc and aγd ∈ RΓc. If
aγd ∈ RΓc2, implies aγc ∈ RΓc2 as aγb ∈ RΓc2 implies a ∈ RΓc, a contradiction to our
supposition. Hence the result follows.
Example 5. Let Z be a Γ-near ring and Γ = {0, 1, 2, 3} consider c = 6 be an non prime
element of Z then ZΓ6 is non prime ideal because 3 6∈ ZΓ6 and 4 6∈ ZΓ6 but 12 ∈ ZΓ6
and 12 6∈ ZΓ62.
In the below proposition, we reverse the situation occurring in lemma 2.
Preposition 3. Suppose that R be Γ-near integral domain and c be a nonzero nonunit
element of R. If c is not a prime element then there exists a ∈ RΓc and b ∈ RΓc such that
aγb ∈ RΓc and aγb ∈ RΓc2.
Proof. Suppose an ideal RΓc is not prime and consider a ∈ RΓc, b ∈ RΓc such that
aγb ∈ RΓc. If the case, aγb 6∈ RΓc2 then for d = (b+ c) ∈ RΓc and aγd ∈ RΓc. Consider
aγd 6∈ RΓc2) implies ac 6∈ RΓc2 and because aγb 6∈ RΓc2 implies a 6∈ RΓc, a contradiction



A. Taouti et al. / Eur. J. Pure Appl. Math, 11 (2) (2018), 449-456 453

to our hypothesis. Hence the result is valid. Supporting the above lemma3 we present the
below example.
Example 6. Let Z8 = {0, 1, 2, 3, 4, 5, 6, 7} and Γ = {0, 2, 4} consider a non-prime element
of Z8 i.e., c = 6 implies 6Z8 = {0, 2, 4}. Consider 6, 4 ∈ 6Z8 such that 6.2.4 = 0 ∈ 6Z8 and
c2 = 62 and 62Z8 = {0, 4}, hence 6.2.4 = 0 ∈ 62Z8. Further we consider 6.4.4 = 4 ∈ 62Z
and take 4, 2 ∈ 6Z8 then 4.2.2 = 0 ∈ 6Z8, and again we get 4.2.2 = 0 ∈ 62Z8, similarly
4.4.2 = 0 ∈ 6Z8 and 4.4.2 = 0 ∈ 62Z8.
Theorem 1. Suppose N be a Γ-near-ring with identity and P be an almost prime ideal
of N . If P is not prime then PΓP = P .
Proof. Let us assume that P ⊆ PΓP . We have to prove that P is prime. Let us suppose
that two ideals A and B contained in N such that AΓB ⊆ P . If AΓB * PΓP then A * P
or B * P . We assume that AΓB * PΓP . Since P * PΓP as a result p ∈ P such that
< p >* PΓP hence (A+ < p >)Γ(B +N) * PΓP . Consider (A+ < p >)Γ(B +N) * P ,
there exist an element a ∈ A, b ∈ B, p0 ∈< p > and q0 ∈ N such that (a+p0)γ(b+q0) 6∈ P
implies aγ(b + q0) 6∈ P , but aγ(b + q0) = aγ(b + q0) − aγb + aγb ∈ P as AΓB ⊆ P , a
contradiction. Hence (A+ < p >)Γ(B +N) ⊆ P implies A ⊆ P .
Corollary 1. Consider N a Γ-near-ring having identity and containing an ideal P . If
PΓP 6= P then P is prime if and only if P is almost prime.
Proposition 4. If P 6= 0 be a proper ideal of a Γ-near-ring N such that P is almost
prime and (PΓP : P ) ⊆ P then P is prime.
Proof. We suppose that P is not a prime ideal of N . Then there exist x/PΓP and y 6∈ P
such that < x > Γ < y >⊆ P . If < x > Γ < y >* PΓP , then the result holds. Hence
< x > Γ < y >⊆ PΓP . Suppose < x > Γ(< y > +P ) ⊆ P . If < x > Γ(< y > +P ) * P
then we have x ∈ P or y ∈ P , a contradiction to our assumption, or else < x > Γ(< y >
+P ) ⊆ PΓP . Thus < x > ΓP ⊆ PΓP implies x ∈ (PΓP : Γ : P ) ⊆ P .
Theorem 2. Suppose N be a Γ-near-ring and let P be an ideal of N . Then the following
statements are equivalent:

i) If elements a, b, c ∈ N with aγ(< b > + < c >) ∈ P and aγ(< b > + < c >) * PΓP
then a ∈ P or b, c in P .

ii) If x ∈ N − P , then (P : Γ :< x > + < y >) = P ∪ (PΓP : Γ :< x > + < y >) for
some y ∈ N .

iii) If x ∈ NP , then (P : Γ :< x > + < y >) = P or (P : Γ :< x > + < y >) = (PγP :
Γ :< x > + < y >) for some y ∈ N .

iv) P is an almost prime.
Proof. (i) implies (ii) Consider t ∈ (P : Γ :< x > + < y >) for some x ∈ N −P , γ ∈ Γ and
y ∈ N . After that tΓ(< x > + < y >) ⊆ P . If tΓ(< x > + < y >) ⊆ PΓP subsequently
t2Γ(PΓP : Γ :< x > + < y >). If tΓ(< x > + < y >* PΓP , then t ∈ P by assumption.
(ii) implies (iii) holds from the truth that if union of two ideal is an ideal then it is equal
to one of them.(iii) implies (iv) Imagine A and B be ideals of N such that AΓB ⊆ P .
Assume A * P and B * P implies a ∈ A and b ∈ B exist with a, b 6∈ P . Now we say that
AΓB * PΓP and consider b1 ∈ B. In that case AΓ(< b > + < b1 >) * P which implies
A ⊆ (P : Γ :< b > + < b1 >). Then by supposition A ⊆ (< b > + < b1 >)ΓPΓP implies
AΓb1 ⊆ PΓP . Consequently AB ⊆ PΓP and therefore P is an almost prime ideal of N .



A. Taouti et al. / Eur. J. Pure Appl. Math, 11 (2) (2018), 449-456 454

(iv) implies (i) is obvious.
Theorem 3. Suppose N1, N2 be any two Γ-near-rings with identity and let P be a proper
ideal of N1. Then P is almost prime if and only if (P × N2) is an almost prime ideal of
N1 ×N2.
Proof. Suppose P be an almost prime ideal of N1 and consider (A1×B1) and (A2×B2) be
ideals of N1×N2 such that (A1×B1)Γ(A2×B2) ⊆ (P ×N2) and (A1×B1)Γ(A2×B2) *
(P ×N2)Γ(P ×N2). In this case (A1ΓA2 ×B1ΓB2) ⊆ (P ×N2) and (A1ΓA2 ×B1ΓB2) *
(PΓP × NΓN),therefore A1ΓA2 × P and A1ΓA2 * PΓP implies A1 ⊆ P or A2 ⊆ P .
Conversely, assume that (P ×N2) is an almost prime ideal of N1×N2 and consider I and
J be ideals of N1 such that IΓJ ⊆ P and IΓJ * PΓP . Then (I×N2)Γ(J×N2) ⊆ (P×N2)
and (I×N2)Γ(J×N2) * (P ×N2)Γ(P ×N2). By hypothesis, we have (I×N2) ⊆ (P ×N2)
or (J ×N2) ⊆ (P ×N2). Thus I ⊆ P or J ⊆ P .
Lemma 3. If c 6= 0 is a nonunit element in Γ-near integral domain R then ideal RΓc is
prime if and only if RΓc is an almost prime.
Proof. Let c 6= 0 is a nonunit element in an Γ-near integral domain R. Assume that ideal
RΓc is an almost prime we need to prove that RΓc is prime. As we know that ideal RΓc
is an almost prime for some a, b ∈ R and aγb ∈ RΓc − RΓcΓRΓc implies either a ∈ RΓc
or b ∈ RΓc where aγb 6∈ RΓcΓRΓc implies aγb ∈ RΓc. Hence RΓc is a prime ideal.
Conversely, suppose that ideal RΓc is prime and we use a result that every prime ideal is
almost prime then RΓc is almost prime ideal which is immediate from Lemma 2.
Lemma 4. Suppose I be an almost prime ideal in a Γ-near integral domain R. Then the
below statements hold.

(i) If element b is a zero divisor in R/I, in that case bΓI ⊆ IΓI.
(ii) If for any ideal J of R such that I ⊆ J where J consists of zero divisors on R/I

then JΓI = IΓI.
(iii) If I is an invertible ideal then I is prime.

Proof. (i) Let us suppose that there is an element c ∈ I such that bγc ∈ I. If b ∈ I then
obviously bΓI ⊆ IΓI, so let b ∈ I. Since we have b 6∈ I, c 6∈ I and bγc ∈ I. Furthermore
I is an almost prime and bγc ∈ IΓI. Also, for any x ∈ I, x + c 6∈ I and bγ(x + c) ∈ I.
Thus, as I is almost prime, bγ(x+ c) ∈ IΓI. As a result bγc ∈ IΓI, bγx ∈ IΓI. Therefore
bΓI ⊆ IΓI. (ii) This is obvious from (i). (iii) Let xγy ∈ I and x ∈ I. Then from (i)
yΓI ⊆ IΓI. Since I is invertible it is immediate that y ∈ I. Thus I is a prime ideal.
Lemma 5. Let S−1I is an almost prime in the ring S−1R, where R be a Γ-near integral
domain. Then I be an almost prime ideal in R and S be a multiplicatively closed subset
of R disjoint from I.
Proof. Suppose for x, y ∈ R and s, t ∈ S, xγy/sγt ∈ S−1(I − IΓI). Then there exists
u,w ∈ S such that uγxγy ∈ I and wγxγy 6∈ IΓI. Therefore, uγxγy ∈ I − IΓI. Since I
is almost prime so uγx ∈ I or y ∈ I. Therefore, either x/s ∈ S−1I or y/t ∈ S−1I implies
S−1I is an almost prime ideal.



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