2-absorbing powerful ideals and related results
H. Ansari-Toroghy
Department of pure Mathematics, Faculty of Mathematical Sciences University of Guilan, Rasht, Iran
F. Farshadifar
Department of Mathematics, Farhangian University, Tehran, Iran
S. Maleki-Roudposhti∗
Department of pure Mathematics, Faculty of Mathematical Sciences University of Guilan, Rasht, Iran
Abstract. LetRbe an integral domain. In this paper, we will introduce the concepts of 2-absorbing powerful (resp. 2-absorbing powerful primary) ideals ofRand obtain some related results. Also, we investigate a submoduleN of anR-moduleMsuch that (N:RM) is a 2-absorbing powerful ideal ofR.
Keywords: Powerful ideal, 2-absorbing powerful ideal, 2-absorbing powerful submodule, 2-absorbing powerful primary ideal.
AMS Mathematical Subject Classification [2010]: 13C13, 13C99.
1. Introduction
1Throughout this paper,R will denote an integral domain with quotient fieldK. Further,Z, Q, andNwill denote respectively the ring of integers, the field of rational numbers, and the set of natural numbers.
The concept of powerful ideals was introduced in [4]. A non-zero ideal I of R is said to be powerful if, wheneverxy∈I for elementsx, y∈K, thenx∈Ror y∈R.
A proper ideal I ofR is said to be strongly prime if, wheneverxy∈I for elementsx, y ∈K, thenx∈I ory∈I[6].
The concept of 2-absorbing ideals was introduced in [3]. A proper idealIofRis a2-absorbing ideal ofRif whenever a, b, c∈Randabc∈I, thenab∈I orac∈Ior bc∈I.
A 2-absorbing ideal I ofR is said to be a strongly 2-absorbing ideal if, wheneverxyz ∈I for elementsx, y, z∈K, then we have eitherxy∈Ior yz∈I orxz ∈I [1].
The purpose of this paper, is to introduce the concepts of 2-absorbing powerful (resp. 2- absorbing powerful primary) ideals of R and study some their basic properties. Moreover, we introduce and investigate the concepts of 2-absorbing powerful (resp. 2-absorbing copowerful) submodule of anR-moduleM.
2. Main results
Definition2.1.We say that a non-zero idealIofRis a2-absorbing powerful idealif, whenever xyz∈I for elementsx, y, z∈K, we have eitherxy∈R oryz∈Ror xz∈R.
Example 2.2. Consider an integral domainZ. ThenK=Qand 2(2/3)(3/4) = 1∈Zimplies thatZis not a 2-absorbing powerful ideal ofZ.
Question 2.3. IfIis a 2-absorbing powerful ideal ofR, is thenIa strongly 2-absorbing ideal ofR?
Theorem 2.4. LetI be an ideal ofR. Then the following statements are equivalent.
∗speaker
1The results in this article are parts of those results appeared in our paper in [2].
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H. Ansari-Toroghy, F. Farshadifar, and S. Maleki-Roudposhti
(a) I is a 2-absorbing powerful ideal ofR.
(b) For each x, y∈K with xy̸∈Rwe have eitherx−1I⊆R ory−1I⊆R.
Example 2.5. Consider an integral domain Z, then K = Q. Let n be a non-zero positive integer number,p1, p2, q1, q2are distinct prime numbers such thatp1, p2̸ |n. Then (p1/q1)(p2/q2)̸∈
Z, (q1/p1)(nZ)̸∈Z, and (q2/p2)(nZ)̸∈Zimplies thatnZis not a 2-absorbing powerful ideal ofZ by Theorem2.4.
Theorem 2.6. LetI be a 2-absorbing powerful ideal ofR. Then we have the following.
(a) If J andH are ideals ofR, thenJ H ⊆I or I2⊆J∪H. (b) If J andI are prime ideals of R, thenJ andI are comparable.
Definition 2.7. We say that a non-zero submodule N of anR-module M is a 2-absorbing powerful submodule ofM if, (N :RM) is a 2-absorbing powerful ideal ofR.
Theorem 2.8. LetI be a 2-absorbing powerful ideal ofRand let T̸=K be an overring of R such thatIT ̸=T, then I2T is a common ideal, andI3T is 2-absorbing powerful in both rings.
Definition 2.9. We say that a non-zero ideal I of R is asemi powerful ideal if, whenever x2∈I for elementx∈K, we havex∈R.
Remark 2.10. Clearly every powerful ideal ofRis a semi powerful ideal ofR. But as we see in the following example the converse is not true in general.
Example 2.11. Consider the integral domain Z. Then K = Q and (4/3)(3/2) = 2 ∈ 2Z implies that 2Zis not a powerful ideal ofZ. But 2Zis a semi powerful ideal ofZ.
Proposition 2.12. (a) IfP is a semi powerful and 2-absorbing powerful ideal ofR, then P is a powerful ideal ofR.
(b) If P1 andP2 are semi powerful ideals ofR, thenP1∩P2 is a semi powerful ideal of R.
Corollary 2.13. Let P be a prime semi powerful 2-absorbing powerful ideal of R. Then P is a strongly 2-absorbing ideal ofR.
Remark 2.14. In view of Proposition 2.12and Corollary 2.13, if R is root closed, then the answer to the question2.3is “Yes”.
Definition 2.15. We say that a 2-absorbing powerful submodule N of anR-moduleM is a minimal 2-absorbing powerful submodule of a submodule H of M, ifH ⊆N and there does not exist a 2-absorbing powerful submoduleT ofM such thatH ⊂T ⊂N.
Theorem 2.16. Let M be a Noetherian R-module. Then M contains a finite number of minimal 2-absorbing powerful submodules.
Definition 2.17. We say that anR-moduleM is a 2-absorbing copowerful if,AnnR(M) is a 2-absorbing powerful ideal ofR.
By a 2-absorbing copowerful submodule of a module we mean a submodule which is a 2- absorbing copowerful module.
Proposition 2.18. Let N be a finitely generated submodule of an R-module M and S be a multiplicatively closed subset ofR. IfN is a 2-absorbing copowerful submodule andAnnR(N)∩S=
∅, thenS−1N is a 2-absorbing copowerful S−1R-submodule ofS−1M.
Proposition 2.19. Let{Ki}i∈I be a chain of strongly 2-absorbing submodules of anR-module M. Then∪i∈IKi is a 2-absorbing copowerful submodule ofM.
Definition 2.20. We say that a 2-absorbing copowerful submoduleN of anR-moduleM is a Maximal 2-absorbing copowerful submodule of a submoduleH of M, ifN ⊆H and there does not exist a 2-absorbing copowerful submoduleT ofM such thatN⊂T ⊂H.
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2-ABSORBING POWERFUL IDEALS AND RELATED RESULTS
Theorem 2.21. Let M be an ArtinianR-module. Then every non-zero submodule ofM has only a finite number of maximal 2-absorbing copowerful submodules.
Definition 2.22. We say that an ideal I of R is a 2-absorbing powerful primary whenever xyz ∈ I for elements x, y, z ∈ K we have either xy ∈ R or (yz)n ∈ R or (xz)m ∈ R for some n, m∈N.
Theorem 2.23. Let Ibe a 2-absorbing powerful primary ideal ofR. Then we have the follow- ing.
(a) If J andH are radical ideals of R, thenJ H ⊆I orI2⊆J∪H.
(b) If J andI are prime ideals of R, thenJ andI are comparable.
Proposition2.24. LetSbe a multiplicatively closed subset ofR. IfIis a 2-absorbing powerful primary ideal ofRsuch thatS∩I=∅, thenS−1Iis a 2-absorbing powerful primary ideal ofS−1R.
References
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3. A. Badawi,On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc.75(2007), 417-429.
4. A. Badawi and E. Houston,Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conducive domains, Commu. Algebra,30(4) (2002), 1591-1606.
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