Throughout this paper, unless otherwise mentioned, S stands for a commutative semigroup.
Definition 2.1. A proper ideal I of a commutative semigroup S is said to be 2-absorbing primary if abc∈ I implies either ab ∈ I or bc∈I or ac∈I for some a,b,c ∈ S.
Since I⊆I for any ideal I of a semigroup S so we have the following result
Theorem 2.2. Let S be a commutative semigroup. Then every 2-absorbing ideal of S is a 2-absorbing primary ideal of S.
The following lemmas are obvious, hence we omit the proof.
Lemma 2.3.([Lemma 2.1, [6]]) Let S be a commutative semigroup. Then every prime ideal of S is a 2-absorbing ideal of S.
Lemma 2.4.([Theorem 2.4, [6]]) Let S be a commutative semigroup. Then every maximal ideal of S is a 2-absorbing ideal of S.
Lemma 2.5.([Lemma 2.9, [6]]) Let P1 and P2 be two prime ideals of a semigroup S. Then P1 ∩ P2 is a 2-absorbing ideal of S.
Corollary 2.6. Let S be a commutative semigroup. Then
(1) if P1 and P2 are two prime ideals of S then P1∩P2 is a 2-absorbing primary ideal of S.
(2) every maximal ideal of S is a 2-absorbing primary ideal of S.
(3) every prime ideal of S is a 2-absorbing primary ideal of S.
Remark 2.7. The following example shows that converse of Lemma 2.3. and 2.4. are not true. Consider the ideal I2={n∈ℕ:n≥2} in the semigroup S=(ℕ∪{0},+), which is 2-absorbing primary (as well 2-absorbing) but neither prime nor a maximal ideal of S.
Remark 2.8. The following example shows that converse of Theorem 2.2 is not true. Consider the ideal I=(m∈ℕ:m≥6) in the semigroup S=(ℕ,+). Then 1+2+3 ∈ I but neither 1+2 ∈ I nor 2+3 ∈ I nor 1+3 ∈ I. Clearly, I is a 2-absorbing primary ideal of S but not a 2-absorbing ideal of S.
A semigroup S is called regular if for each element s ∈ S there exists an element x ∈ S such that sxs=s ([7], Section 5). Since in a commutative regular semigroup every ideal coincide with its radical ([7], Theorem 5.1), we have the following result.
Corollary 2.9. Let S be a commutative regular semigroup. Then an ideal I of S is a 2-absorbing primary ideal of S if and only if I is a 2-absorbing ideal of S.
The following is a characterization of a semigroup in which 2-absorbing primary ideals are prime:
Theorem 2.10. Let S be a commutative semigroup. Then every 2-absorbing primary ideals of S are prime if and only if prime ideals of S are linearly ordered and A=A for every 2-absorbing primary ideal A of S.
Proof. Let P1 and P2 be two prime ideals of S. Then P1∩P2 is a 2-absorbing primary ideal of S (cf. Corollary 2.6(1)) and so prime by hypothesis. Hence prime ideals are linearly ordered. Again let A be a 2-absorbing primary ideal of S and so prime ideal of S. Therefore A=A.
Conversely, Let A be a 2-absorbing primary ideal of S. Since prime ideals are linearly ordered so A=A=∩ α∈ΛPα=Pβ for some β∈Λ, where {Pα}α∈Λ are prime ideals of S containing A. Hence the result follows.
Since every primary ideals of a commutative semigroup S is 2-absorbing primary, we have the following result by using Theorem 3.1 of [12].
Corollary 2.11. Let S be a commutative semigroup with zero and identity in which nonzero 2-absorbing primary ideals are prime. Then S satisfies one of the following conditions.
(i) S=H∪M, where H is the group of units in S and M={0,ah:a∈M,a2=0,h∈H}.
(ii) Mn=M for every positive integer n.
The following is a characterization of a semigroup in which 2-absorbing primary ideals are maximal:
Theorem 2.12. Let S be a commutative semigroup with unity. Then 2-absorbing primary ideals of S are maximal if and only if S is either a group or has a unique 2-absorbing primary ideals A such that S=A∪H, where H is the group of units of S.
Proof. Let S be commutative semigroup with unity in which 2-absorbing primary ideals are maximal. If S is not a group, it has unique maximal ideal say A and since maximal ideals are 2-absorbing primary (cf. Corollary 2.6(2)) so has unique 2-absorbing primary ideal A. Therefore S=A ∪ H, where H is the group of units of S.
Conversely, if S is a group then it has no 2-absorbing primary ideal so the condition satisfied vacously. Again if S has unique 2-absorbing primary ideal then clearly it is maximal.
Moreover, we prove that A is also a group. Clearly A is the unique prime ideal of S. Then for any a ∈ A, aS=A. Hence aS is a 2-absorbing primary ideal of S. Hence aS=A for every a ∈ A, by hypothesis. Then aS=a2S=A implies a=a2x⇒ax=a2x2. Thus ax is an idempotent element of A. If possible let e, f be two idempotent element of S. Then eS=fS⇒e=fe=ef=f. Consequently eS=aS=A. Therefore A is a group. So we can conclude the corollary
Corollary 2.13. Let S be a commutative semigroup with unity. Then 2-absorbing primary ideals are maximal if and only if either S is a group or S is a union of two groups.
Theorem 2.14. Let S be a commutative semigroup. Then every semiprimary ideal of S is a 2-absorbing primary ideal of S.
Proof. Let I be a semiprimary ideal of a semigroup S and abc∈I⊆I with ab∉I for some a,b,c∈S. Hence I is a prime ideal of S.
Case(1). Suppose ab∉I. Since I is a prime ideal of S so c∈I. Hence ac∈I and bc∈I.
Case(2). Suppose ab∈I. Since I is a prime ideal, we have either a∈I or b∈I. Hence either ac∈I or bc∈I.
Therefore I is a 2-absorbing primary ideals of S.
The following are obvious consequence of above theorem:
Corollary 2.15. Let I be an ideal of a commutative semigroup S. Then
(1) if I is a prime ideal of S, then I is a 2-absorbing primary ideal of S.
(2) if I is a prime ideal of S, then In is a 2-absorbing primary ideal of S for each natural number n.
(3) every primary ideal of S is a 2-absorbing primary ideal of S.
Remark 2.16. The converse of Theorem 2.14 is not true. Consider the principal ideal I=(6) generated by 6 in the semigroup S={ℤ,.}, which is clearly 2-absorbing primary but I=(6) is not a prime ideal of S and hence not a semiprimary ideal of S.
The following theorem is a characterization of a semigroup in which 2-absorbing primary ideals are semiprimary:
Theorem 2.17. Let S be a commutative semigroup. Then the following statements are equivalent:
(1) 2-absorbing primary ideals of S are semiprimary.
(2) Prime ideals of S are linearly ordered.
(3) S is a semiprimary semigroup.
(4) Semiprime ideals are linearly ordered.
(5) Semiprime ideals of S are prime.
Proof. (1)⇒(2) Let P1 and P2 be two prime ideals of S. Then P1∩P2 is a 2-absorbing primary ideal of S (cf. Corollary 2.6(1)) and so semiprimary ideal of S, by hypothesis. Therefore P1∩P2=P1∩P2=P1∩P2, is a prime ideal of S. Therefore either P1⊆P2 or P2⊆P1.
(2)⇒(1) Since prime ideals of S are linearly ordered, then for any ideal I of S, I is a prime ideal of S. Consequently, 2-absorbing primary ideals of S are semiprimary.
(2)⇔(3) follows from ([9], Theorem 1).
(2)⇒(4) Let S1 and S2 are two distinct semiprime ideals of S. Then S1∩S2 is a semiprime ideal of S. Hence S1∩S2=S1∩S2, is a prime ideal of S, since prime ideals are linearly ordered. Hence semiprime ideals of S are linearly ordered.
(4)⇒(2) It is clear.
(2)⇒(5) Let I be a semiprime ideal of S. Then I=I, is a prime ideal of S.
(5)⇒(2) Let P1 and P2 be two distinct prime ideals of S. Then P1∩P2 is a semiprime ideals of S, hence prime ideals of S. Consequently, prime ideals of S are linearly ordered.
Definition 2.18. A commutative semigroup S is said to be 2-absorbing primary if every proper ideal of S is a 2-absorbing primary ideal of S.
Example 2.19. Consider the commuttive semigroup S={ℕ,+}, which has no proper prime ideal. Clearly S is a primary semigroup and hence 2-absorbing primary semigroup.
Theorem 2.20. Let S be a commutative semigroup with unity. If
(i) proper prime ideals are maximal then S is a 2-absorbing primary semigroup.
(ii) 2-absobing ideals are semiprimary then S is a 2-absorbing primary semigroup.
Proof. (i) If S is commutative semigroup with unity in which proper prime ideals are maximal. Then S has unique proper prime as well as maximal ideal say M. Clearly for any ideal I of S, I=M, a prime ideal of S. Hence S is a 2-absorbing primary semigroup (cf. Corollary 2.15(1)).
(ii) Let P1 and P2 be two prime ideals of S. Then P1∩P2 is 2-absorbing ideal of S and hence semiprimary ideal of S. Then P1∩P2=P1∩P2=P1∩P2, is a prime ideal of S and hence prime ideals of S are linearly ordered. Then for any ideal I of S, I is a prime ideal of S and hence S is a 2-absorbing primary ideal of S (cf. Corollary 2.15(1)).
Theorem 2.21. Let S be a commutative semigroup. Then S can be written as disjoint union of 2-absorbing primary semigroup.
Proof. Let S be a commutative semigroup. Then S can be written as disjount union of archimedian subsemigroups of S ([12], Corollary 1.6). Clearly every archimedian semigroup is primary and every primary semigroup is 2-absorbing primary. Therefore S=∪Sα, where Sα is a 2-absorbing primary subsemigroup of S.
Theorem 2.22. Let S be a commutative regular semigroup. Then the following statements about S are equivalent:
(1) 2-absorbing primary ideals of S are semiprimary.
(2) prime ideals of S are linearly ordered.
(3) idempotents of S form a chain under natural ordering.
(4) All ideals of S are linearly ordered.
(5) S is a fully prime semigroup.
(6) S is a primary semigroup.
(7) S is a semiprimary semigroup.
(8) 2-absorbing ideals of S are prime.
Proof. (1) ⇒ (2) follows from Theorem 2.17.
(2)⇒(3)⇒(4)⇒(5)⇒(6)⇒(7) follows from ([9], Theorem 1).
(7) ⇒ (8) Let I be a 2-absorbing ideal of S. Since S is semiprimary and regular so I=I=P, a prime ideal of S, as desired.
(8)⇒(1) Let I be a 2-absorbing primary ideal of a regular semigroup S. Then I=I is a 2-absorbing ideal of S (cf. Theorem 2.24(1)) and hence prime by hypothesis. Therefore I is a semiprimary ideal of S, as desired.
Lemma 2.23. Let I be a 2-absorbing ideal of a semigroup S with unity. Then (I:x) is a 2-absorbing ideal of S for all x ∈ S-I.
Proof. Let abc ∈ (I:x) for some a,b,c ∈ S. Since x ∉ I, we have (I:x)=S. Then abcx ∈ I. Since I is a 2-absorbing ideal of S so either ab ∈ I or bcx∈I or acx ∈ I that is ab ∈ (I:x) or ac ∈ (I:x) or bc ∈ (I:x). Consequently (I:x) is a 2-absorbing ideal of S.
Theorem 2.24. Let I be a 2-absorbing primary ideal of a semigroup S with unity. Then the following statements are true
(1) I is a 2-absorbing ideal of S.
(2) If I is a prime ideal P of S then (I:x) is a 2-absorbing primary ideal of S with (I:x)=P for all x∈S−I.
(3) If I is a maximal ideal M of S then (I:x) is a 2-absorbing primary ideal of S with (I:x)=M for all x∈S−I.
(4) (I:x) is a 2-absorbing ideal of S for all x∈S−I.
(5) (I:x)=(I:x2) for all x∈S−I.
Proof. (1) Proof is similar to ([3], Theorem 2.2).
(2) Let x∈S−I and p∈(I:x). Then px∈I⊆I. Since I is a prime ideal of S and x∉I so p∈I. Hence I⊆(I:x)⊆I=P, which implies P=I⊆(I:x)⊆I=P. Consequently, (I:x)=P, a prime ideal of S and hence (I:x) is a 2-absorbing primary ideal of S (cf. Corollary 2.15.(1)).
(3) If I=M and maximal ideal of a semigroup with unity is prime, hence the proof follows from (2).
(4) Since I is a 2-absorbing primary ideal of S so I is a 2-absorbing ideal of S. Hence (I:x) is a 2-absorbing ideal of S (cf. Lemma 2.23).
(5) Let p∈(I:x2). Then x2p∈I and hence either x2∈I or xp∈I, since I is a 2-absorbing ideal of S. If x2∈I then x∈I, a contradiction. Hence xp∈I implies p∈(I:x). Therefore (I:x2)⊆(I:x). Clearly, (I:x)⊆(I:x2). Hence, (I:x)=(I:x2) for all x∈S−I.
Theorem 2.25. Let I be an ideal of a semigroup S. Then I is a 2-absorbing primary ideal of S if and only if I is a 2-absorbing ideal of S.
Proof. Suppose I is a 2-absorbing primary ideal of S and abc∈I for some a,b,c∈S. So either ab∈I or bc∈I=I or ca∈I=I. Hence I is a 2-absorbing ideal of S.
Conversely, let I be a 2-absorbing ideal of S. Since 2-absorbing ideals are 2-absorbing primary (cf. Theorem 2.2) so clearly I is a 2-absorbing primary ideal of S.
Clearly arbitary union of 2-absorbing primary ideals of a semigroup S is 2-absorbing primary ideal but intersection of two 2-absorbing primary ideals need not be 2-absorbing primary ideal of S. We have the following example
Example 2.26. Let I1=5ℤ and I2=6ℤ be two ideals of the semigroup (ℤ,.). Clearly I1 and I2 are 2-absorbing primary ideal of S. Then I1∩I2=30ℤ, which is not a 2-absorbing ideal of S. Hence I1∩I2 is a not a 2-absorbing primary ideal of S (cf. Theorem 2.24(1)).
Let M be the set of all 2-absorbing primary ideals of a semigroup S and we define a relation ρ on M by I1ρI2 if and only if I1=A=I2 for some 2-absorbing ideal A of S. Clearly ρ is a congrurence on M. So every element of a ρ-equivalence class A is a A-2-absorbing primary ideal for some 2-absorbing ideal A of S. Clearly M forms a semigroup with respect to usual set union and each ρ-equivalence class of M is an element of the factor semigroup M/ρ.
Theorem 2.27. Each ρ-class of M is closed under finte intersections.
Proof. Let I1,I2,....In be elements of a ρ-class A for some 2-absorbing ideal A of S. Then (I1∩I2∩.....∩In)=A. Let abc∈I=∩Ii and ab ∉ I for some a,b,c ∈ S. Then ab ∉ Ii for some i∈{1,2,....,n}. Hence bc∈Ii=A or ac∈Ii=A. Hence the result follows.
Therefore the semigroup M can be written as disjoint union of semigroups that is M=∪{A:A∈M/ρ}.
Theoprerm 2.28. Let I1 and I2 be two P1-primary and P2- primary ideals for some prime ideals P1 and P2 of a commutative a semigroup S. Then I1∩I2 and I1I2 are 2-absorbing primary ideals of S.
Proof. The proof is similar to ([3], Theorem 2.4).
Proposition 2.29. Let I be a proper ideal of a semigroup S. Then the following statements are equivalent
(1). For every ideals J, K, L of S such that I⊆J, JKL⊆I implies JK⊆I or KL⊆I or JL⊆I.
(2). For every ideals J,K,L of S, JKL⊆I implies either JK⊆I or KL⊆I or JL⊆I.
Proof. (1)⇒(2) Let JKL⊆I for some ideals J, K, L of S. Then (J∪I)KL=JKL∪IKL⊆I. Setting P=I∪J we have I⊆P and PKL⊆I implies either PK⊆I or PL⊆I or KL⊆I. Therefore either (J∪I)K⊆I or (J∪I)L⊆I or KL⊆I implies JK⊆I or KL⊆I or JL⊆I, as desired.
(2)⇒(1) Straightforward.
Let S[x] be the polynomial semigroup of a semigroup S. Then if I is an ideal of S, then I[x]={axi:a∈I,i≥0} is an ideal of S[x] and also we have the following result:
Lemma 2.30. Let I be an ideal of a semigroup S. Then I[x]=I[x].
Proof. Let f(x)=axi∈I[x] for some a ∈ S and i≥ 0. Then for some n∈ℕ, anxin∈I[x]⇒an∈I⇒a∈I⇒f(x)=axi∈I[x]. Therefore I[x]⊆I[x].
Again let g(x)=bxi∈I[x]. Then b∈I⇒bn∈I for some n∈ℕ. Hence (g(x))n=bnxin∈I[x]⇒g(x)∈I[x]. Therefore I[x]⊆I[x]. Consequently, I[x]=I[x].
Proposition 2.31. Let S be a commutative semigroup. Then a proper ideal I is a 2-absorbing primary ideal of S if and only if I[x] is a 2-absorbing primary ideal of S[x].
Proof.,Let I be a 2-absorbing primary ideal of S and (axi)(bxj)(cxk)∈I[x], where a,b,c ∈ S and i,j,k ≥ 0. Then abcxi+j+k∈I[x]⇒abc∈I⇒ab∈I or ac∈I or bc∈I, since I is a 2-absorbing primary ideal of S. Hence (axi)(bxj)=abxi+j∈I[x] or (axi)(cxj)=acxi+j∈I[x]=I[x] or (bxj)(cxk)=bcxj+k∈I[x]=I[x], since I[x]=I[x] (cf. Lemma 2.30). Therefore I[x] is a 2-absorbing primary ideal of S[x].
Conversely, let I[x] be a 2-absorbing primary of S[x] and abc ∈ I for some a,b,c ∈ S. Then (axi)(bxj)(cxk)=abcxi+j+k∈I[x], for some i,j,k≥0. Since I[x]=I[x] (cf. Lemma 2,30) so ab∈ I or ac∈I or bc∈I. Therefore I is a 2-absorbing primary ideal of S.
The following result is a simple consequence of Proposition 2.31:
Corollary 2.32. Let I be a proper ideal of S. Then I is a 2-absorbing primary ideal of S if and only if I[x1,x2,…,xn] is a 2-absorbing primary ideal of S[x1,x2,…,xn].
Let C be a non-empty subsemigroup of a commutative semigroup S. Let ρ be the relation defined on S×C by (x,a)ρ(y,b) if and only if cay=cxy for some c∈ C. Clearly ρ is a congrurence on S×C. Then C−1S={sc:s∈S,c∈C} is the quoitent semigroup of (S×C) modulo {ρ}, called the semigroup of fractions.
The composition on C−1S is defined as (xa)(yb)=xyab. Also if I is an ideal of S, then C−1I={ic:i∈I,c∈C} is an ideal of C−1S, moreover C−1I=C−1I.
Theorem 2.33. Let C be a subsemigroup of a semigroup S and I be an ideal of S such that C∩I=ϕ. If I is a 2-absorbing primary ideal of S then C−1I is a 2-absorbing primary ideal of C−1S.
proof. Let I be a 2-absorbing primary ideal of S and (as)(br)(ct)∈C−1I for some a,b,c ∈ S and s,r,t ∈ C. Hence there exists some p ∈ C such that abcp ∈ I. Therefore either ab ∈ I or bcp∈I or acp∈I. Now ab ∈ I implies (as)(br)=absr∈C−1I, bcp∈I implies (br)(ct)=bcprtp∈C−1I=C−1I and acp∈I implies (as)(ct)=acpstp∈C−1I=C−1I, consequently, C−1I is a 2-absorbing primary ideal of C−1S.
Lemma 2.34. Let f:S→S′ be a homomorphism of semigroups. Then the following statements holds
(1) f−1( I′ )=f−1( I ′), where I' is an ideal of S'.
(2) If f is an isomorphism, then f(I)=f(I), where I is an ideal of S.
Theorem 2.35. Let f:S→S′ be a homomorphism of semigroups. Then the following statements holds:
(1) If I' is a 2-absorbing primary ideal of S', then f−1(I′) is a 2-absorbing primary ideal of S.
(2) Let I be a proper ideal of S such that {(x,y)∈kerf:x=y}⊆I×I. Then
(i) If f(I) is a 2-absorbing primary ideal of S', then I is a 2-absorbing primary ideal of S.
(ii) If f is onto and I is a 2-absorbing primary ideal of S, then f(I) is a 2-absorbing primary ideal of S'.
(3) If f is an isomorphism and I is a 2-absorbing primary ideal of S, then f(I) is a 2-absorbing primary ideal of S'.
Proof. (1) The proof is similar as that of ([3],Theorem 2.20(1)).
(2) The proof of (i) and (ii) are similar as that of Theorem 2.18 of [4] by replacing 2-absorbing ideal I as 2-absorbing primary ideal of S and then use the result of (1).
(3) It is trivial.
As a simple consequence of above theorem, we have the following result
Corollary 2.36. (1) Let S⊆S′ be an extension of semigroup S and I be a 2-absorbing primary ideal of S'. Then I∩S is a 2-absorbing primary ideal of S. (2) Let I⊆J be two ideals of S. Then J is a 2-absorbing primary ideal of S if and only if J/ I is a 2-absorbing primary ideal of S/ I.
Lemma 2.37. Let S=S1×S2, where each Si is a semigroup. Then the following statement holds
(1) If I1 is an ideal of S1, then I1×S2=I1×S2.
(2) If I2 is an ideal of S2, then S1×I2=S1×I2. (3) I1×I2=I1×I2.
Theorem 2.38. Let S=S1×S2, where each Si is a semigroup. Then the following statements holds
(1) I1 is a 2-absorbing primary ideal of S1 if and only if I1×S2 is a 2-absorbing primary ideal of S.
(2) I2 is a 2-absorbing primary ideal of S2 if and only if S1×I2 is a 2-absorbing primary ideal of S.
Proof. (1) Suppose I1×S2 is a 2-absorbing primary ideals of S. Let abc∈I1 for some a,b,c∈S1. Then (abc,x3)∈I1×S2 for some x∈S2. So (a,x)(b,x)(c,x)∈I1×S2 and hence either (a,x)(b,x)∈I1×S2 or (b,x)(c,x)∈I1×S2 or (a,x)(c,x)∈I1×S2 and so either ab∈I1 or bc∈I1 or ac∈I1, which implies I1 is a 2-absorbing primary ideal of S1.
Conversely, Suppose I1 is a 2-absorbing primary ideal of S1. Let (a,x)(b,x)(c,x)∈I1×S2 for some a,b,c ∈ I1 and x ∈ S. Then abc∈I1 implies ab ∈ I1 or bc∈I1 or ac∈I1 and hence either (a,x)(b,x)∈I1×S2 or (b,x)(c,x)∈I1×S2 or (a,x)(c,x)∈I1×S2, which implies I1×S2 is a 2-absorbing primary ideal of S1×S2.
(2) The proof is similar to (1).
Theorem 2.39. Let S=S1×S2, where each Si is a semigroup. Let I1 and I2 are ideals of S1 and S2 respectively. If I=I1×I2 is a 2-absorbing primary ideal of S then I1 and I2 are 2-absorbing primary ideals of S1 and S2 respectively.
Proof. Let abc∈I1 for some a,b,c∈ S. Then (a,x)(b,x)(c,x)=(abc,x3)∈I1×I2 for some x ∈ I2. Since I1×I2 is a 2-absorbing primary ideal of S, either (a,x)(b,x)=(ab,x2)∈I1×I2 or (b,x)(c,x)=(bc,x2)∈I1×I2=I1×I2 or (a,x)(c,x)=(ac,x2)∈I1×I2=I1×I2 (cf. Lemma 2.37(3)). Hence ab∈I1 or bc∈I1 or ac∈I1. Consequently I1 is a 2-absorbing primary ideal of S1. Similarly, we can prove that I2 is a 2-absorbing primary ideal of S2.
Remark 2.40. The following example shows that converse of Theorem 2.39 is not true. Consider the 2-absorbing primary ideals I1=6ℤ and I2=5ℤ of the semigroup (ℤ,.). Then (2,3)(3,2)(5,5)∈I1×I2 but nether (2,3)(3,2)∈I1×I2 nor (2,3)(5,5)∈I1×I2 nor (3,2)(5,5)∈I1×I2. Hence I1×I2 is not a 2-absorbing primary ideal of ℤ×ℤ.
Theorem 2.41. Let S=S1×S2, where S1, S2 are commutative semigroup with zero and identity. Let I be a proper ideal of S. Then the following statements are equivalent
(1) I is a 2-absorbing primary ideal of S.
(2) Either I=I1×S2 for some 2-absorbing primary ideal I1 of S1 or I=S1×I2 for some 2-absorbing primary ideal I2 of S2 or I=I1×I2 for some primary ideals I1 of S1 and I2 of S2 respectively.
Proof. The proof is similar to ([14], Theorem 2.17).