/compile/output.dvi


On 2-Absorbing Primary Ideals in Commutative Semirings

Pratibha Kumar1, Manish Kant Dubey2, Poonam Sarohe 3,∗

1 Department of Mathematics, Kirori Mal College, University of Delhi, Delhi 110007, India.
2 SAG, DRDO, Metcalf House, Delhi 110054, India.
3 Department of Mathematics, Lakshmibai College, University of Delhi, Delhi 110052, India.

Abstract. In this paper, we define 2-absorbing and weakly 2-absorbing primary ideals in a commutative

semiring S with 1 ≠ 0 which are generalization of primary ideals of commutative ring. A proper ideal

I of a commutative semiring S is said to be a 2-absorbing primary (weakly 2-absorbing primary) ideal

of S if abc ∈ I (0 ≠ abc ∈ I) implies ab ∈ I or bc ∈
√

I or ac ∈
√

I . Some results concerning 2-

absorbing primary and weakly 2-absorbing primary ideals are given. It is proved that a subtractive

weakly 2-absorbing primary ideal I that is not a 2-absorbing primary ideal satisfies
√

I =
√

0.

2010 Mathematics Subject Classifications: 16Y30, 16Y60

Key Words and Phrases: Semiring, Subtractive ideal, 2-absorbing primary ideal, Weakly 2-absorbing

primary ideal, Q-ideal

1. Introduction

The algebraic structure of semiring plays a prominent role in various branches of mathe-

matics as well as some other branches of applied science. The concept of semiring was first

introduced by H. S. Vandiver [13] in 1934 and has since then been studied by many authors.

The structure of prime ideals in semiring theory have gained importance and many mathe-

maticians have exploited its usefulness in algebraic systems over the decades. Anderson and

Smith [2] introduced the notion of weakly prime ideals in commutative ring for the study of

factorization in commutative rings with zero divisors. The concepts of 2-absorbing and weakly

2-absorbing ideals of commutative ring with nonzero unity have been introduced by Badawi

[7] and Badawi and Darani [8] respectively which are generalizations of prime and weakly

prime ideals in commutative rings. Recently, Badawi et al. [9] introduced the concept of 2-

absorbing primary ideals in commutative rings with 1 ≠ 0 and gave some characterizations

related to it.

∗Corresponding author.

Email addresses: pratibhakumar313@gmail.com (P. Kumar), kantmanish@yahoo.com (M. K. Dubey),

poonamsarohe@gmail.com(Poonam Sarohe)

http://www.ejpam.com 186 © 2016 EJPAM All rights reserved.

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 2, 2016, 186-195

ISSN 1307-5543 – www.ejpam.com



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 187

A commutative semiring is a commutative semigroup (S, ⋅) and a commutative monoid

(S,+, 0S) in which 0S is the additive identity and 0S ⋅ x = x ⋅ 0S = 0S for all x ∈ S, both are

connected by ring like distributivity. A nonempty subset I of a semiring S is called an ideal of

S if a, b ∈ I and r ∈ S, a + b ∈ I and ra, ar ∈ I . An ideal I of a semiring S is called subtractive

if a, a + b ∈ I , b ∈ S, then b ∈ I .

Let I be an ideal of S. Then, the radical of I is defined as

Rad(I) =
√

I = {a ∈ S ∶ an ∈ I for some positive integer.}n

Annihilator of a semiring S is defined as Ann(a) = {x ∈ S ∶ ax = 0}. Recall from [10],

that a proper ideal I of a commutative semiring S is said to be a 2-absorbing (weakly 2-

absorbing)ideal of S if whenever a, b, c ∈ S and abc ∈ I (0 ≠ abc ∈ I), then ab ∈ I or ac ∈
I or bc ∈ I . It is easy to see that every 2-absorbing ideal of a semiring S is a weakly 2-

absorbing ideal of S but converse need not be true. For further understanding the concept

of semiring, refer [11] and the properties of a 2-absorbing and weakly 2-absorbing ideals in

commutative semirings, we refer[10]. The paper is organized as follows: In section 2, we

introduce the concepts of 2-absorbing primary ideal of a commutative semiring and prove

some results corresponding to ring theory. In section 3, we introduce the concept of weakly 2-

absorbing primary ideal of a commutative semiring and give some generalizations of [5, 6, 10]

and [12] which are analogous to commutative ring theory. Throughout this paper, semiring S

is considered as commutative with identity 1 ≠ 0.

2. 2-Absorbing Primary Ideals

In this section, we introduce the concept of 2-absorbing primary ideal of a commutative

semiring and prove some results related to it.

Definition 1. Let S be a commutative semiring and I be a proper ideal of S. Then I is said to be

a 2-absorbing primary ideal of S if whenever a, b, c ∈ S and abc ∈ I , then ab ∈ I or ac ∈
√

I or

bc ∈
√

I .

It is easy to see that every 2-absorbing ideal of a commutative semiring S is a 2-absorbing

primary ideal of S but converse need not be true. For instance, consider a semiring S = Z+∪{0}
and an ideal I = ⟨8⟩ of S. Then I is a 2-absorbing primary ideal of S but it is not a 2-absorbing

ideal of S, as 2.2.2 ∈ ⟨8⟩ but 2.2 ∉ ⟨8⟩. Also, every primary ideal of S is a 2-absorbing primary

ideal of S but converse is not true, as ⟨10⟩ is a 2-absorbing primary ideal of S but it is not a

primary ideal of S.

Theorem 1. Let f ∶ S ↦ S′ be a homomorphism of commutative semirings. Then, if I ′ is a

2-absorbing primary ideal of S′, then f −1(I ′) is a 2-absorbing primary ideal of S.

Proof. Let abc ∈ f −1(I ′) for some a, b, c ∈ S. Then f (abc) ∈ I ′, that is, f (a) f (b) f (c) ∈ I ′.

Since I ′ is a 2-absorbing primary ideal of S′, therefore f (a) f (b) ∈ I ′ or f (b) f (c) ∈
√

I ′ or

f (c) f (a) ∈
√

I ′. Hence, ab ∈ f −1(I ′) or bc ∈ f −1(
√

I ′) or ca ∈ f −1(
√

I ′). Since

f −1(
√

I ′) ⊆
√

f −1(I ′), we have f −1(I ′) is a 2-absorbing primary ideal of S.



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 188

Theorem 2. If I is a 2-absorbing primary ideal of a semiring S, then
√

I is a 2-absorbing ideal

of S.

Proof. Let abc ∈
√

I for some a, b, c ∈ S. Suppose that ac ∉
√

I and bc ∉
√

I . Since

abc ∈
√

I , then there exists a positive integer n such that (abc)n = an bncn ∈ I . This gives

an bn ∈ I , since I is a 2-absorbing primary ideal of S and ac ∉
√

I and bc ∉
√

I . Hence, ab ∈
√

I .

Thus,
√

I is a 2-absorbing ideal of S.

Corollary 1. Let I be an ideal of a semiring S. Then the following statements are equivalent:

(1) I is a 2-absorbing primary ideal of S.

(2)
√

I is a 2-absorbing ideal of S and if abc ∈ I with bc ∉
√

I and ca ∉
√

I then ab ∈ I .

Definition 2. Let I be a 2-absorbing primary ideal of a semiring S. Then by above theorem

P =
√

I is a 2-absorbing ideal of S. In this case, I is said to be a P − 2-absorbing primary ideal of

S.

Theorem 3. Let I1, I2, . . . , In be P − 2-absorbing primary ideals of S, where P is a 2-absorbing

ideal of S. Then I =
n

⋂
i=i

Ii is a P − 2−absorbing primary ideal of S.

Proof. Proof is similar to [9, Theorem 2.16].

Theorem 4. Let S be a semiring. Suppose that I1 is a P1−primary ideal of S for some prime

ideal P1 of S, and I2 is a P2−primary ideal of S for some prime ideal P2 of S. Then the following

statements hold:

(1) I1 I2 is a 2-absorbing primary ideal of S.

(2) I1 ∩ I2 is a 2-absorbing primary ideal of S.

Proof. Proof is similar to [9, Theorem 2.4].

Theorem 5. Let I be a 2-absorbing primary ideal of S such that
√

I = P is a prime ideal of S.

Then (I ∶ x) is a 2-absorbing primary ideal of S with
√
(I ∶ x) = P for all x ∈ S ∖

√
I , where

(I ∶ x) = {r ∈ S ∶ x r ∈ I}.

Proof. Let x ∈ S ∖
√

I and a ∈ (I ∶ x). Then ax ∈ I ⊆
√

I , gives a ∈
√

I , since x ∉
√

I and
√

I

is prime. Hence, a ∈
√

I , gives I ⊆ (I ∶ x) ⊆
√

I = P, which implies that

P =
√

I ⊆
√
(I ∶ x) ⊆

√
I = P. Thus, we have

√
(I ∶ x) = P. Now, let a, b, c ∈ S be such that

abc ∈ (I ∶ x). Then abcx ∈ I , implies that either abc ∈ I or ax ∈
√

I or bcx ∈
√

I . If ax ∈
√

I

or bcx ∈
√

I , we get ac ∈
√
(I ∶ x) or bc ∈

√
(I ∶ x), since

√
(I ∶ x) =

√
I and x ∉

√
I . Next, if

abc ∈ I , we have either ab ∈ I or bc ∈
√

I or ca ∈
√

I , since I is a 2-absorbing primary ideal

of S. Thus, ab ∈ (I ∶ x) or bc ∈
√
(I ∶ x) or ca ∈

√
(I ∶ x). Therefore (I ∶ x) is a 2-absorbing

primary ideal of S.



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 189

Theorem 6. If I is a 2-absorbing primary ideal of a semiring S, then the following holds:

(1) (
√

I ∶ x) is a 2-absorbing ideal of S for all x ∈ S ∖
√

I .

(2) (
√

I ∶ x) = (
√

I ∶ x2) for all x ∈ S ∖
√

I .

Proof. (1) Let a, b, c ∈ S be such that abc ∈ (
√

I ∶ x). Then abcx ∈
√

I . Since
√

I is a

2-absorbing ideal of S therefore ab ∈
√

I or bcx ∈
√

I or cax ∈
√

I , that is, ab ∈ (
√

I ∶ x) or

bc ∈ (
√

I ∶ x) or ca ∈ (
√

I ∶ x). Hence (
√

I ∶ x) is a 2-absorbing ideal of S.

(2) It is clear that (
√

I ∶ x) ⊆ (
√

I ∶ x2). Let y ∈ (
√

I ∶ x2). Then x2 y ∈
√

I . Since√
I is 2-absorbing ideal of S, therefore we have either x2 ∈

√
I or x y ∈

√
I . If x y ∈

√
I ,

then y ∈ (
√

I ∶ x) and we are done. If x2 ∈
√

I , then x ∈
√

I , a contradiction. Hence,

(
√

I ∶ x) = (
√

I ∶ x2).
Let S be a semiring and A be the set of all multiplicatively cancellable elements of S (so

1 ∈ S). For further understanding of the structure of the semiring of fractions SA of S with

respect to A, refer [3].

Theorem 7. Let I be a 2-absorbing primary ideal of a semiring S and A be the multiplicatively

cancellable subset of S. Then ISA is a 2-absorbing primary ideal of SA.

Proof. Let a/s, b/t, c/r ∈ SA, where a, b, c ∈ S and s, t, r ∈ A be such that abc/st r ∈ ISA but

bc/t r ∉
√

ISA and ca/rs ∉
√

ISA. Then there exist p ∈ I and z ∈ A such that abcz = st rp ∈ I

but bcz ∉ I and caz ∉ I since if bcz ∈ I and caz ∈ I , we get bc/t r ∈
√

ISA and ca/rs ∈
√

ISA,

which leads to a contradiction. Since abcz ∈ I and I is a 2-absorbing primary ideal of S, we

have ab ∈ I , implies ab/st ∈ ISA. Hence, ISA is a 2-absorbing primary ideal of SA.

Lemma 1. Let I be a 2-absorbing primary ideal of S. Suppose that I and
√

I be subtractive ideals

of S and abJ ⊆ I for some a, b ∈ S and an ideal J of S. If ab ∉ I , then either aJ ⊆
√

I or bJ ⊆
√

I .

Proof. Suppose that aJ /⊆
√

I and bJ /⊆
√

I . Therefore, there are some x , y ∈ J such that

ax ∉
√

I and b y ∉
√

I . Since abx ∈ I and ab ∉ I and ax ∉
√

I , we have bx ∈
√

I . Since ab y ∈ I

and ab ∉ I and b y ∉
√

I , we have a y ∈
√

I . Now, since ab(x + y) ∈ I and ab ∉ I , we have

a(x + y) ∈
√

I or b(x + y) ∈
√

I , since I is a 2-absorbing primary ideal of S. If a(x + y) ∈
√

I

and a y ∈
√

I , then ax ∈
√

I , since
√

I is subtractive, which is a contradiction. Similarly, if

b(x + y) ∈
√

I and bx ∈
√

I , we get b y ∈
√

I , a contradiction. Hence, either aJ ⊆
√

I or

bJ ⊆
√

I .

Theorem 8. Let I be a proper subtractive ideal of S and suppose that
√

I is a subtractive ideal of

S. Then I is a 2-absorbing primary ideal of S if and only if whenever I1 I2 I3 ⊆ I for some ideals

I1, I2, I3 of S, then either I1 I2 ⊆ I or I2 I3 ⊆
√

I or I3 I1 ⊆
√

I .

Proof. Proof is similar to the proof of [9, Theorem 2.19]

Definition 3 ([1, Definition (4)]). An ideal I of a semiring S is called a Q-ideal (partitioning

ideal) if there exists a subset Q of S such that



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 190

(1) S = ∪{q + I ∶ q ∈ Q}

(2) If q1, q2 ∈ Q, then (q1 + I) ∩ (q2 + I) ≠ ∅ ⇔ q1 = q2.

Let I be a Q-ideal of a semiring S. Then S/IQ = {q + I ∶ q ∈ Q} forms a semiring under

the following addition ‘⊕’ and multiplication ’⊙’, ( q1 + I) ⊕ (q2 + I) = q3 + I , where q3 ∈ Q

is unique such that q1 + q2 + I ⊆ q3 + I and (q1 + I) ⊙ (q2 + I) = q4 + I , where q4 ∈ Q is

unique such that q1q2 + I ⊆ q4 + I . This semiring S/IQ is called the quotient semiring of S and

denoted by (S/IQ,⊕,⊙) or S/IQ. By definition of Q-ideal, there exists a unique q0 ∈ Q such

that 0+ I ⊆ q0 + I . Then q0 + I is a zero element of S/IQ. Clearly, if S is commutative then S/IQ
is commutative.

Theorem 9. Let S be a semiring, I a Q-ideal of S and P a subtractive ideal of S such that I ⊆ P.

Then P is a 2-absorbing primary ideal of S if and only if P/IQ∩P is a 2-absorbing primary ideal

of S/IQ.

Proof. Let P be a 2-absorbing primary ideal of S. Suppose that q1 + I , q2 + I , q3 + I ∈ S/IQ
are such that (q1 + I) ⊙ (q2 + I) ⊙ (q3 + I) = q4 + I ∈ P/IQ∩P where q4 ∈ Q ∩ P is a unique

element such that q1q2q3 + I ⊆ q4 + I ∈ P/IQ∩P . So q1q2q3 = q4 + i for some i ∈ I . Since P is a 2-

absorbing primary ideal of S and q1q2q3 ∈ P, therefore q1q2 ∈ P or (q2q3)m ∈ P or (q3q1)n ∈ P

for some positive integers m, n. Consider the case q1q2 ∈ P. If (q1 + I) ⊙ (q2 + I) = i1 + I

where i1 ∈ Q is a unique element such that q1q2 + I ⊆ i1 + I . So i1 + f = q1q2 + e for some

e, f ∈ I . Since P is subtractive and I ⊆ P, we have i1 ∈ P, therefore i1 ∈ Q ∩ P. Thus, P/IQ∩P

is a 2-absorbing primary ideal of S/IQ. Next, if qm
2 qm

3 ∈ P for some positive integer m. Let

(qm
2 + I) ⊙ (qm

3 + I) = i2 + I where i2 ∈ Q is a unique element such that qm
2 qm

3 + I ⊆ i2 + I .

So, i2 + f1 = qm
2 qm

3 + e1 for some f1, e1 ∈ I . Since P is subtractive and I ⊆ P, we have i2 ∈ P,

therefore i2 ∈ Q ∩ P. This gives,

(q2 + I)m ⊙ (q3 + I)m = (qm
2 + I)⊙ (qm

3 + I) = qm
2 qm

3 + I ⊆ i2 + I

where i2 ∈ Q ∩ P. Hence, P/IQ∩P is a 2-absorbing primary ideal of S/IQ. Similarly, if (q3q1)n ∈ P

for some positive integer n, we get P/IQ∩P is a 2-absorbing primary ideal of S/IQ.

Conversely, if P/IQ∩P is a 2-absorbing primary ideal of S/IQ. Let abc ∈ P for some a, b, c ∈ S.

Since I is a Q-ideal of S therefore there exist q1, q2, q3, q4 ∈ Q such that a ∈ q1 + I , b ∈ q2 + I ,

c ∈ q3 + I . Now, abc ∈ (q1 + I) ⊙ (q2 + I) ⊙ (q3 + I) = q4 + I . So, abc = q4 + i3 ∈ P for some

i3 ∈ I . Since P is a subtractive ideal of S and I ⊆ P, we have q4 ∈ P. So,

(q1 + I)⊙ (q2 + I)⊙ (q3 + I) = q4 + I ∈ P/IQ∩P ,

which gives (q1+I)⊙(q2+I) ∈ P/IQ∩P or (qr
2+I)⊙(qr

3+I) ∈ P/IQ∩P or (qt
3+I)⊙(qt

1+I) ∈ P/IQ∩P

for some positive integers r, t, since P/IQ∩P is a 2-absorbing primary ideal of S/IQ. If (q1+ I)⊙
(q2+I) ∈ P/IQ∩P , then there exists q5 ∈ Q ∩ P such that ab ∈ (q1+I)⊙(q2+I) = q5+I . This gives,

ab = q5 + i4 for some i4 ∈ I . This implies ab ∈ P. Thus P is a 2-absorbing primary ideal of S. If

(qr
2+I)⊙(qr

3+I) ∈ P/IQ∩P , then there exists q6 ∈ Q ∩ P such that br cr ∈ (qr
2+I)⊙(qr

3+I) = q6+I .

This gives, br cr = q6+i5 for some i5 ∈ I . This implies, (bc)r ∈ P. Therefore, bc ∈
√

P. Similarly,

we can prove that ca ∈
√

P. Hence, P is a 2-absorbing primary ideal of S.



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 191

3. Weakly 2-Absorbing Primary Ideals

In this section, we introduce the notion of weakly 2-absorbing primary ideal of a commu-

tative semiring and prove some results related to it.

Definition 4. Let S be a commutative semiring and I be a proper ideal of S. Then I is said to be

a weakly 2-absorbing primary ideal of S if whenever a, b, c ∈ S and 0 ≠ abc ∈ I , then ab ∈ I or

ac ∈
√

I or bc ∈
√

I .

It is clear that every 2-absorbing primary ideal of S is a weakly 2-absorbing primary ideal

of S but converse is not true, as ⟨0⟩ is a weakly 2-absorbing primary ideal of S but not a 2-

absorbing primary ideal of S. Consider the set S = Z16 = {0, 1, 2, . . . , 15}. Then S forms a

semiring under addition and multiplication modulo 16. If we take the set I = {0, 8}. Then it is

easy to check that I is a weakly 2-absorbing primary ideal of S but it not a weakly 2-absorbing

ideal of S because 0 ≠ 2.2.2 ∈ I but 2.2 ∉ I . For any ideal, the following implications hold:

Prime ⇒ 2-absorbing ⇒ Weakly 2-absorbing

ideal /⇐ ideal /⇐ ideal

	⇓ 	⇓ 	⇓
Primary ⇒ 2-absorbing primary ⇒ Weakly 2-absorbing primary

ideal /⇐ ideal /⇐ ideal

Lemma 2 ([6, Lemma 2.5]). Let I be a subtractive ideal of a semiring S and let a ∈ I and

a + b ∈
√

I . Then b ∈
√

I .

Proof. Let a ∈ I and a + b ∈
√

I . Then, we can assume that there exists a positive integer m

such that (a + b)m = c + bm ∈ I , where c ∈ I (as a ∈ I). This gives bm ∈ I since I is subtractive.

Hence b ∈
√

I .

Theorem 10. Let S be a semiring and I be a subtractive weakly 2-absorbing primary ideal that

is not a 2-absorbing primary ideal of S. Then
√

I =
√

0.

Proof. We first prove that I3 = 0. Suppose that I3 ≠ 0. Then, we prove that I is a 2-

absorbing primary ideal of S. Let abc ∈ I for some a, b, c ∈ S. Suppose that abc ≠ 0, then

ab ∈ I or bc ∈
√

I or ac ∈
√

I since I is a weakly 2-absorbing primary ideal of S. So, assume

that abc = 0. If abI ≠ 0, then there exists an element a′ in I such that aba′ ≠ 0, which implies

0 ≠ aba′ = ab(c + a′) ∈ I . Since I is a weakly 2-absorbing primary ideal of S, therefore either

ab ∈ I or b(c + a′) ∈
√

I or a(c + a′) ∈
√

I . By Lemma 2, we have ab ∈ I or bc ∈
√

I or

ac ∈
√

I . So, we assume that abI = 0. Similarly, we can assume that aIc = 0 and I bc = 0.

Now, let aI2 ≠ 0. Then there exist i1, i2 ∈ I such that ai1i2 ≠ 0. Since abI = aIc = I bc = 0,

we have 0 ≠ a(b + i1)(c + i2) = ai1i2 ∈ I . Therefore, either a(b + i1) ∈ I or a(c + i2) ∈
√

I or

(b + i1)(c + i2) ∈
√

I . Hence, we have either ab ∈ I or ac ∈
√

I or bc ∈
√

I . So, we can assume

that aI2 = 0. Likewise, we can assume that bI2 = 0 and cI2 = 0. Since I3 ≠ 0, there exist

p, q, r ∈ I such that pqr ≠ 0. Again, (a + p)(b + q)(c + r) = pqr ∈ I , so either (a + p)(b + q) ∈ I

or (b+q)(c+ r) ∈
√

I or (a+p)(c+ r) ∈
√

I , that is, ab+aq+pb+pq ∈ I or bc+br+qc+qr ∈
√

I



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 192

or ac + ar + pc + pr ∈
√

I . Hence, either ab ∈ I or bc ∈
√

I or ac ∈
√

I . This implies that I is a

2-absorbing primary ideal of S, which is a contradiction. Therefore, I3 = 0. Clearly,
√

0 ⊆
√

I .

As I3 = 0, we get I ⊆
√

0. This concludes that
√

I ⊆
√

0. Thus,
√

I =
√

0.

Theorem 11. Let S be a semiring and {Ii}i∈∆ be a family of subtractive weakly 2-absorbing

primary ideals of S that are not 2-absorbing primary ideals of S. Then I = ⋂
i∈∆

Ii is a weakly

2-absorbing primary ideal of S.

Proof. Let {Ii}i∈∆ be a family of weakly 2-absorbing primary ideals of S that are not 2-

absorbing primary ideals of S. Therefore, by Theorem 10, we have
√

Ii =
√

0 for all i ∈ ∆.

This gives ⋂
i∈∆

√
Ii =
√

0. Thus we have
√

I =
√

0, since ⋂
i∈∆

√
Ii =
√

I . Next, let a, b, c ∈ S be

such that 0 ≠ abc ∈ I but ab ∉ I . Then there exists i ∈ ∆ such that ab ∉ Ii and 0 ≠ abc ∈ Ii .

This gives bc ∈ √Ii or ac ∈ √Ii since Ii is a weakly 2-absorbing primary ideal of S and ab ∉ Ii .

Thus, either bc ∈ √Ii =
√

0 =
√

I or ca ∈ √Ii =
√

0 =
√

I . Hence I is a weakly 2-absorbing

primary ideal of S.

Definition 5 ([4, Definition 1(i)]). A proper ideal I of a semiring S is said to be a strong ideal,

if for each a ∈ I there exists b ∈ I such that a + b = 0.

Proposition 1. Let S and S′ be semirings, f ∶ S ↦ S′ be an epimorphism such that f (0) = 0 and

I be a subtractive strong ideal of S. Then the following holds:

(1) If I is a weakly 2-absorbing primary ideal of S such that ker f ⊆ I , then f (I) is a weakly

2-absorbing primary ideal of S′.

(2) If I is a 2-absorbing primary ideal of S such that ker f ⊆ I , then f (I) is a 2-absorbing

primary ideal of S′.

Proof. (1) Let a, b, c ∈ S′ be such that 0 ≠ abc ∈ f (I). Then there exists an element m ∈ I

such that 0 ≠ abc = f (m). Since f is an epimorphism, therefore there exist p, q, r ∈ S such

that f (p) = a, f (q) = b, f (r) = c. Also, since I is a strong ideal of S and m ∈ I , therefore there

exists n ∈ I such that m + n = 0. This implies f (n + m) = 0, that is, f (pqr + n) = 0, implies

pqr + n ∈ ker f ⊆ I . So, 0 ≠ pqr ∈ I (as I is a subtractive ideal of S) because if pqr = 0, then

f (m) = 0, a contradiction. Since I is a weakly 2-absorbing primary ideal of S, therefore either

pq ∈ I or qr ∈
√

I or rp ∈
√

I . Thus ab ∈ f (I) or bc ∈ f (
√

I) ⊆
√

f (I) or ac ∈ f (
√

I) ⊆
√

f (I).
Hence, f (I) is a weakly 2-absorbing primary ideal of S′.

(2) It follows from (1).

Proposition 2. Let a, x ∈ S. Then the following holds:

(1) suppose Sx be a subtractive ideal S and if Ann(x) ⊆ Sx. Then Sx is a 2-absorbing primary

ideal of S if and only if Sx is a weakly 2-absorbing primary ideal of S.

(2) suppose aI be a subtractive ideal S and if Ann(a) ⊆ aI. Then aI is a 2-absorbing primary

ideal of S if and only if it is a weakly 2-absorbing primary ideal of S.



P. Kumar, M. Dubey, P. Sarohe / Eur. J. Pure Appl. Math, 9 (2016), 186-195 193

Proof. (1) Let Sx be a weakly 2-absorbing primary ideal of S and r, s, t ∈ S with rst ∈ Sx . If

rst ≠ 0, then rs ∈ Sx or r t ∈
√

Sx or st ∈
√

Sx , which implies Sx is a 2-absorbing primary ideal

of S. So we assume that rst = 0. Evidently, rs(x + t) ∈ Sx . If rs(x + t) ≠ 0, we have rs ∈ Sx

or r(x + t) ∈
√

Sx or s(x + t) ∈
√

Sx , as Sx is a weakly 2-absorbing primary ideal of S. By

Lemma 2, we have either rs ∈ Sx or r t ∈
√

Sx or st ∈
√

Sx . Therefore, we have rs(x + t) = 0

implies rsx = 0 and so rs ∈ Ann(x) ⊆ Sx and thus rs ∈ Sx . Hence Sx is a 2-absorbing primary

ideal of S.

(2) Let aI be a weakly 2-absorbing primary ideal and r, s, t ∈ S such that rst ∈ aI . If rst ≠ 0

then rs ∈ aI or r t ∈
√

aI or st ∈
√

aI , which implies aI is a 2-absorbing primary ideal of S.

So, we assume rst = 0. Clearly, r(s + a)t = rst + rat ∈ aI . If r(s + a)t ≠ 0, then r(s + a) ∈ aI

or r t ∈
√

aI or (s + a)t ∈
√

aI . By Lemma 2, we get either rs ∈ aI or r t ∈
√

aI or st ∈
√

aI .

So, we assume that r(s + a)t = 0 implies rat = 0, as rst = 0. Hence r t ∈ Ann(a) ⊆ aI . Thus

r t ∈ aI and hence aI is a 2-absorbing primary ideal of S.

Consider S = S1 × S2 where each Si , i = 1, 2 is a commutative semiring with unity and

(a1, a2)(b1, b2) = (a1 b1, a2 b2) for all a1, b1 ∈ S1 and a2, b2 ∈ S2.

Proposition 3. Let I be a proper ideal of a semiring S1. Then the following statements are

equivalent:

(1) I is a 2-absorbing primary ideal of S1.

(2) I × S2 is a 2-absorbing primary ideal of S = S1 × S2.

(3) I × S2 is a weakly 2-absorbing primary ideal of S = S1 × S2.

Proof. (1) ⇒ (2) Let (a1, a2), (b1, b2), (c1, c2) ∈ S be such that

(a1, a2)(b1, b2)(c1, c2) ∈ I × S2. Then (a1 b1c1, a2 b2c2) ∈ I × S2 implies a1 b1c1 ∈ I . This gives

either a1 b1 ∈ I or (b1c1)m ∈ I or (a1c1)n ∈ I for some positive integers m, n, since I is a 2-

absorbing primary ideal of S1. If a1 b1 ∈ I , then (a1, a2)(b1, b2) ∈ I × S2. If bm
1 cm

1 ∈ I for some

positive integer m, then (bm
1 , bm

2 )(cm
1 , cm

2 ) ∈ I ×S2, that is, (bm
1 cm

1 , bm
2 cm

2 ) ∈ I ×S2. Similarly, we

can prove the case when (a1c1)n ∈ I for some positive integer n. Hence, I ×S2 is a 2-absorbing

primary ideal of S.

(2) ⇒ (3) It is obvious.

(3) ⇒ (1) Let abc ∈ I for some a, b, c ∈ S1. Then for each 0 ≠ r ∈ S2, we have

(0, 0) ≠ (a, 1)(b, 1)(c, r) ∈ I × S2. This gives (a, 1)(b, 1) ∈ I × S2 or (bm, 1)(cm, rm) ∈ I × S2

or (cn, rn)(an, 1) ∈ I × S2, since I × S2 is a weakly 2-absorbing primary ideal of S. That is,

either ab ∈ I or bmcm ∈ I or ancn ∈ I for some positive integers m, n. This shows that I is a

2-absorbing primary ideal of S1.

Theorem 12. Let (S, M) be a local semiring with M3 = 0. Then every proper subtractive ideal

of S is a weakly 2-absorbing primary ideal of S.

Proof. Proof is analogous to the proof of [10, Theorem 2.8].



REFERENCES 194

Theorem 13. Let S be a semiring, I a Q-ideal of S and P a subtractive ideal of S such that I ⊆ P.

Then

(1) if P is a weakly 2-absorbing primary ideal of S, then P/I(Q∩P) is a weakly 2-absorbing

primary ideal of S/I(Q).

(2) if I and P/I(Q∩P) are weakly 2-absorbing primary ideals of S and S/I(Q) respectively, then

P is a weakly 2-absorbing primary ideal of S.

Proof. (1) If ( q1 + I)⊙ (q2 + I)⊙ (q3 + I) ≠ 0 in S/IQ then q1q2q3 ≠ 0 in S, then the proof

follows from Theorem 9.

(2) Let a, b, c ∈ S be such that 0 ≠ abc ∈ P. If abc ∈ I , then either ab ∈ I ⊆ P or

bc ∈ I ⊆
√

I ⊆
√

P or ca ∈ I ⊆
√

I ⊆
√

P, since I is a weakly 2-absorbing primary ideal of

S. So, assume that abc ∉ I . Then there are elements q1, q2, q3 ∈ Q such that a ∈ q1 + I ,

b ∈ q2 + I , c ∈ q3 + I . Therefore, for some i1, i2, i3 ∈ I , a = q1 + i1, b = q2 + i2, c = q3 + i3.

As abc = q1q2q3 + q1q2i3 + q1q3i2 + q1i2i3 + q2q3i1 + q2i1i3 + q3i1i2 + i1i2i3 ∈ P and since P is

subtractive, we have q1q2q3 ∈ P. Consider, (q1 + I)⊙ (q2 + I)⊙ (q3 + I) = q4 + I where q4 is the

unique element such that q1q2q3 + I ⊆ q4 + I . Since P is subtractive, we have q4 ∈ P ∩Q, hence

q1q2q3+ I ⊆ q4+ I ∈ P/IQ∩P , that is, (q1+ I)⊙(q2+ I)⊙(q3+ I) ∈ P/IQ∩P . Let q ∈ Q be the unique

element such that q+ I is the zero element in S/IQ. If (q1+ I)⊙(q2+ I)⊙(q3+ I) = 0S/IQ = q+ I ,

then there exit r, s ∈ I such that q1q2q3 + r = q + s ∈ I . Therefore, q1q2q3 ∈ I , since I is a Q-

ideal of S, it is subtractive by [Cor. 8.23, 11]. This gives abc ∈ I , a contradiction. Hence,

0S/IQ ≠ (q1 + I) ⊙ (q2 + I) ⊙ (q3 + I) ∈ P/IQ∩P . This gives either (q1 + I) ⊙ (q2 + I) ∈ P/IQ∩P

or (ql
2 + I) ⊙ (ql

3 + I) ∈ P/IQ∩P or (qt
3 + I) ⊙ (qt

1 + I) ∈ P/IQ∩P for some positive integers l, t

since P/IQ∩P is a weakly 2-absorbing primary ideal of S/IQ. Thus, either ab ∈ P or (bc)l ∈ P

or (ca)t ∈ P for some positive integers l, t. Hence, P is a weakly 2-absorbing primary ideal of

S.

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