Abstract
The concepts of weakly 2-absorbing ideal and weakly 1-absorbing prime ideal in an almost distributive lattice (ADL) are introduced, and the necessary conditions for a weakly 1-absorbing prime ideal to become a weakly 2-absorbing ideal in algebraic form are proved. Also, weakly 2-absorbing ideals are characterized in terms of weakly prime ideals and 2-absorbing ideals. Finally, the lattice epimorphic images and inverse images of the weakly 2-absorbing ideal and weakly 1-absorbing prime ideal are discussed.
1. Introduction
Badawi [1] was introduced the concept of 2-absorbing ideals on a commutative ring and assume that all rings are commutative with , which is a generalization of prime ideals and some properties of these were studied. Subsequently, several researchers worked on 2-absorbing ideals in different cases (refer to [2–5]). Later in the paper [6], Yassine et al. obtained on 1-absorbing prime ideals of a commutative ring. In the papers [7–10] (Figures 1and 2), the concept of weakly prime ideals were introduced and some of its properties are investigated. The concept of an almost distributive lattice (ADL) was introduced by Swamy and Rao [11] (Figures 3 and 4) as a common abstraction to most of the existing ring theoretic generalizations of a Boolean algebra and which is an algebra of type satisfying all the axioms of a distributive lattice with zero except commutative, commutative, and right distributivity of over . The concept of prime ideal is a vital role in the study of structure theory of distributive lattices in general and of Boolean algebras in particular, see [11].
In this paper, we introduce the concept of 2-absorbing ideal in an almost distributive lattice which is a generalization of prime ideals in an ADL. A proper ideal of an ADL is called a 2-absorbing ideal (2-AI for short) of if whenever and , then or or . It is shown that a proper ideal of is a -AI of if and only if whenever for some ideals of , for , then or or . In addition to this, it is observed that the lattice epimorphic image and inverse image of 2-AI and -AI are also 2-AI and -AI, respectively. Next, we introduce the concepts of weakly 2-AI of an ADL which is weaker than that of weakly prime ideal and 2-AI of an ADL. The Cartesian product of weakly 2-AIs is also discussed here, and some equivalent conditions for the set of all weakly 2-AIs to become 2-AIs under the Cartesian product are derived. Mainly, we have proved that all prime ideals are 2-AIs and weakly prime ideals, that weakly prime ideals are weakly 2-AIs, and also that all 2-AIs are weakly 2-AIs and vice versa is not true; examples were given to shown these. It is further demonstrated that all prime ideals are 1-absorbing prime ideals, that 1-absorbing prime ideals are 2-AIs, and that all weakly prime ideals and 1-absorbing prime ideals are weakly 1-absorbing prime ideals and that weakly 1-absorbing prime ideals are weakly 2-AIs, and there are examples that show that the converse of these is not true. Finally, it is also shown that the image and inverse image of weakly 2-AI, 1-absorbing prime ideal, and weakly 1-absorbing prime ideal under lattice epimorphism are again weakly 2-AI, 1-absorbing prime ideal, and weakly 1-absorbing prime ideal, respectively.
Throughout this paper, stands for an ADL with a maximal element and stands for a complete lattice satisfying the infinite meet distributive law, that is, for any and .
2. Preliminaries
In this section, we recall certain definitions, results, and notations which will be needed later on are presented, see [1, 11, 12].
Definition 1. An algebra of type is called an ADL if it satisfies the following conditions for all and (1)(2)(3)(4)(5)(6)
Each of the axioms (1) through (6) above is independent from the others. The element is called the zero element.
Any bounded below distributive lattice is an ADL.
Example 1. Let be a nonempty set. Fix an arbitrary element . For any , define and on by,
Then is an ADL with as its zero element. This ADL is called the discrete ADL.
Several ring theoretic generalizations of Boolean algebras (other than Boolean rings which are precisely Boolean algebras) can be made as an ADL. The following example is one such.
Example 2. Let be a commutative regular ring with identity (that is, is a commutative ring with unity in which, for each , there exists an (unique) idempotent such that ). For any , define
Then, is an ADL with the additive identity as the zero element.
Theorem 2. Let be an ADL. For any and , we have (1) and (2)(3)(4)(5)(6)(7)(8)
Definition 3. Let be an ADL. For any and , define if (equivalently ).
Then is a partial order on .
Theorem 4. The following hold good for any elements , and of an ADL . (1)(2) and (3)(4) (i.e., is associative on )(5)(6)The set is a bounded distributive lattice under the induced operations and with as the smallest element and as the largest element(7) whenever (8)(9) and (10)(11)
Theorem 5. For any elements and of an ADL , the following are equivalent to each other. (1)(2)(3)(4)(5) exists in and is equal to (6)There exists such that and (7) exists in is equal to
Theorem 6. The following statements are equivalent for any ADL . (1) for all (2) for all (3) is a distributive lattice bounded below(4) for all (5)(i.e., ) for all (6) (i.e., ) for all (7)For any
As a consequence, for any ideal of for all and An element is said to be maximal if, for any implies . It can be easily observed that is maximal if and only if for all .
Definition 7. Let be a nonempty subset of an ADL Then, is called an ideal of if and for all .
Definition 8. Let be an ADL and for any subset of , let
Then, is the smallest ideal of containing and is called the ideal generated by in . Also,
When , then we simply write for and call this the principal ideal generated by in . The principal ideal generated by in is given by
Theorem 9. Let be an ADL and and . Then, the following holds good. (1)(2)(3) and
Theorem 10. Let be an ADL and and be ideals of . Then, in the lattice , , and Also, the lattice is distributive.
Definition 11. A nonzero proper ideal of is called a 2-absorbing ideal of if for any and , then or or .
3. 2-Absorbing Ideal
In this section, we introduce the notion of 2-absorbing ideal (2-AI) and -absorbing (-AI) of a given almost distributive lattice (ADL) and prove several structural properties of these.
Definition 12. Let be an ADL. A proper ideal of is said to be a -absorbing ideal of and denoted by -AI if for any
Theorem 13. Let be a -AI of . For such that , we have
Proof. For such that . Then
(by 2.7(4))
. (again by 2.7(4))
Since is a -AI of and , hence . Thus Similarly, or .
Lemma 14. Let and be ideals of and be a -AI of . The following assertions hold for any . (1) or or (2) or or
Proof. Let be a -AI of and and ideals of . (1)Suppose . Let . Then, . Since is a 2-AI of , if whenever and then or or . From this, we have . Hence, . Suppose and . Now, (by given); this a gives a contradiction. Thus, or (2)Suppose Let such that and . It follows that ; that is, , a contradiction. Therefore, or
Theorem 15. Let be a proper ideal of . The following statements are equivalent: (1) is a -AI of (2)For ideals of , or or (3)For ideals of , or or
Proof. : suppose is a 2-absorbing ideal of . Let , for some proper ideals , and of . Let such that and and put . It follows that, and . By Lemma 14(2), , we get either or . If , then (by Theorem 5(4)), for all , which implies that , so contradiction. Similarly, if , then (by 2.7(4)), for all , implies that , so contradiction. Thus, either or . Conversely suppose implies that either or or , for any ideals of . Let , we have . Suppose also that and . Let and . Since , it follows that and . Then by the above lemma, ; that is, . Thus, is a 2-absorbing ideal of .
(2) and are clear
Definition 16. Let and be ADLs and form the set by . Define and in by, and , for any .
Then, is an ADL under the pointwise operations and is the zero element in .
Let us recall from [11] that a proper ideal of is said to be a prime ideal if, for any and either or . Now, we have the following.
Theorem 17. Every prime ideal of is a -AI of .
Proof. Assume that is a prime ideal of . Let , . Then, either or , or or , and hence (since is an ideal and by 2.10). If , then it is obvious and if , and . Thus, is a -AI of .
The following example show that the converse of Theorem 44 is not true.
Example 3. Let be a discrete ADL with as its zero element defined in Example1and be the lattice represented by the Hasse diagram given below:
Consider . Then, is an ADL (note that is not a lattice) under the pointwise operations and on and , the zero element in . Then, is a 2-AI of but is an ideal which is not prime, since , for all .
Theorem 18. Let and be prime ideals of . Then, is a -AI of .
Proof. Let and . Then, and . Since and are prime ideals of , we have either or , or or , or or (and or , or or , or or ). Suppose that or and or . If and , then . If and , then either and or and . Hence the theorem.
Definition 19. Let and be ADLs. A mapping is called a lattice homomorphism if the following are satisfied, for any . (1)(2)(3)
Theorem 20. Let and be ADLs and be a lattice homomorphism. Then, the following holds. (1)If is an epimorphism and is a 2-AI of , then is a AI of (2)If is an isomorphism and is a 2-AI of , then is a AI of
Proof. Let and be ADLs and be a lattice homomorphism. (1)Let be a -AI of and . Let . Then, . Since is a -AI of , . Thus, . Now if and , then and and hence . Thus, . Therefore, is a -AI of (2)Let such that and , for some . As , we have . Since is a -AI of , implies that either or or . That is either or or . Thus, is a -AI of
Theorem 21. Let and be ADLs. If is a -AI of , then is a -AI of . Also, if is a -AI of , then is a -AI of .
Proof. Let be a -AI of and such that , for every . Then, . Since is a -AI of , we have either or or . So that, either or or , for every . Thus, is a -AI of . Similarly, is a -AI of .
Definition 22. A proper ideal of is a weakly prime ideal of if for any ,
Lemma 23. Every prime ideal of is a weakly prime ideal of .
Proof. It is clear.
The converse of the above lemma is not true; consider the following example.
Example 4. Let be an ADL discussed in Example 3. Let . Clearly, is a weakly prime ideal of , while is not a prime ideal of , since implies that and , for all . Thus, every weakly prime ideal of is not a prime ideal of .
Definition 24. A proper ideal of is a weakly -AI of if for any ,
Lemma 25. Every weakly prime ideal of is a weakly -AI of .
Proof. It is clear.
The following example show that the converse of Lemma 25 is not true.
Example 5. Let be a discrete ADL with as its zero element defined in Example1and be the lattice represented by the Hasse diagram given below.
Consider . Then, is an ADL (which is not a lattice) under the pointwise operations and on and , the zero element in . Let . Then, implies and , for all . Thus, is a weakly -AI of . But is neither prime ideal nor weakly prime ideal of , since and .
Theorem 26. Every -AI of is a weakly -AI of .
Proof. It is clear.
The following example show that the converse of Theorem 26 is not true.
Example 6. Let be a discrete ADL with as its zero element defined in Example1 and be a lattice whose Hasse diagram is given below.
Consider . Then, is an ADL (which is not a lattice) under the pointwise operations and on and , the zero element in . Let . Clearly is a weakly -AI of . On the other hand, consider which implies that , and , for all . Thus is not a -AI of . Therefore, every weakly -AI of is not a -AI of .
As a consequence of Theorem 18 and Lemmas 23 and 25, we have the following.
Theorem 27. Let and be weakly prime ideals of . Then, the intersection of and is also a weakly -AI of .
Theorem 28. Let be a proper ideal of and in . Then, is a -AI of if and only if is a weakly -AI of .
Proof. It is clear.
As a consequence of Theorems 20 and 26, we have the following.
Theorem 29. Let be a lattice homomorphism. Then, the following holds. (1)If is an epimorphism and is a weakly AI of , then is a weakly 2-AI of (2)If is an isomorphism and is a weakly AI of , then is a weakly 2-AI of
Theorem 30. Let be an ADL, where and are ADLs. Let and be proper ideal of and , respectively. If is a weakly -AI of , then and are weakly -AI of and , respectively.
Proof. Suppose that is a weakly -AI of . Let and such that . Then, implies that either or or . From this, either or or . Thus, is a weakly -AI of . Similarly, is a weakly -AI of .
The converse of the above theorem is not true, consider the following example.
Example 7. Let be the lattice discussed in Example 5 and be a chain represented by the diagram given below.
Consider . Let and be ideals of and , respectively. Then, . We note that, for all , . Now, , and . It follows that is not a weakly -AI of .
Theorem 31. Let be ADL, where and be ADLs and be a proper ideal of . Then, the following are equivalent. (1) is a weakly -AI of (2) is a -AI of (3) is a -AI of
Proof. assume (1). Let such that , for every . Then, , implies either or or , for every (since is a weakly -AI of ). Thus, is a -AI of .
assume (2). Let such that . Since is a -AI of , , for every , which implies that either or or . From this, we have that either or or . Therefore, is a -AI of .
Suppose is a -AI of and , for every and . Then, , implies either or or , for every (since either or or ). Thus, is a weakly -AI of .
Theorem 32. Let be ADL, where and be ADLs. Let and be proper ideal of and , respectively. Then, the following are equivalent. (1) is a weakly -AI of (2) and is a -AI of or is a prime ideal of and is a prime ideal of (3) is a -AI of
Proof. assume (1). If , then is a -AI of (by the above theorem). Suppose that . Let such that and let . Then, . Since , we conclude that either or and hence either or . Thus, is a prime ideal of . Similarly, is a prime ideal of .
: assume (2). Then, by the above theorem, is a -AI of . Suppose that is a prime ideal of and is a prime ideal of . Then, clearly is a prime ideal of . Let such that . Then, either or , or or , or or . Thus, is a -AI of .
(3) is clear by Theorem 26
In the following, we introduce the concept of -AI of an ADL .
Definition 33. Let be a proper ideal of and . Then, is an -AI of if whenever , for , , then there are of the whose meet is in .
Corollary 34. Let be a proper ideal of and . Then, (1) is -AI if and only if whenever , for with , then there are of the whose meet is in (2)If is -AI, then is an -AI, for all
Corollary 35. Let be a lattice homomorphism. Then, the following holds. (1)If is an epimorphism and is an -AI of , then is an -AI of (2)If is an isomorphism and is an -AI of , then is an -AI of
Theorem 36. If is a nonempty chain of -AI of , then is an -AI of .
Proof. Let and such that . Let and , for all . Then, for each , there exist an -AI such that . Assume that . Let . If , then , for each . Since and is -AI of , we have . Again, and is -AI of , then . So, , for every . Thus, . Hence the theorem.
4. 1-Absorbing Prime Ideals
In this section, we introduce the 1-absorbing prime ideals of ADLs.
Definition 37. Let be an ADL. A proper ideal of is a 1-absorbing prime ideal if for such that , then either or .
Theorem 38. Let be a proper ideal of . Then, every prime ideal of is a 1-absorbing prime ideal and every 1-absorbing prime ideal of is a 2-absorbing ideal of .
The following example show that every 2-AI of is not 1-absorbing prime ideal of .
Example 8. Let be an ADL discussed in Example 6. Let . Then, for all ,
Thus, is a 2-AI of . On the other hand, implies and . From this, is not a 1-absorbing prime ideal of . Therefore, every 2-AI of is not 1-absorbing prime ideal of .
Next, we have the following result.
Theorem 39. Let be an ADL where and be ADLs with , proper ideal of and a proper ideal of . Then, is a 1-absorbing prime ideal of if and only if or , where and are prime ideals of and , respectively.
Lemma 40. Let be a 1-absorbing prime ideal of . If , for all proper ideal of and for , then or .
Theorem 41. Let be a proper ideal of . Then, the following are equivalent. (1) is a -absorbing prime ideal of (2)If for some proper ideals , and of , then either or
Proof. : suppose is a 1-absorbing prime ideal of . Let for some proper ideals , and of . Let . Then, there exists and such that and hence . Since , it follows that (by the above lemma).
: assume (2) hold. Suppose that , for and let . Suppose also that and . Then and (by assumption, ). Thus by the above lemma, , that is, and thus . Therefore, is a 1-absorbing prime ideal of .
Theorem 42. Let and be ADLs and be a lattice homomorphism. Then, the following hold. (1)If is a 1-absorbing prime ideal of , then is a 1-absorbing prime ideal of (2)If is onto and is a 1-absorbing prime ideal of with then is a 1-absorbing prime ideal of
Proof. (1)Let be a 1-absorbing prime ideal of and for some . Then which implies that or . It follows that or . Hence is a 1-absorbing prime ideal of (2)Let be a 1-absorbing prime ideal of with , be onto and for some . Since is onto, then there exists such that and . Therefore, . Since we conclude that . Thus, or and so, or . Therefore, is a 1-absorbing prime ideal of
Definition 43. Let be a proper ideal of . Then, is said to be a weakly 1-absorbing prime ideal of if for some , then either or .
Theorem 44. Every weakly prime ideal of is a weakly 1-absorbing prime ideal of .
Proof. It is clear.
The following example show that the converse of Theorem 44 is not true.
Example 9. Let be an ADL discussed in Example 5. Let . Let . Then,
Thus, is a weakly 1-absorbing prime ideal of , but is not weakly prime ideal of , since implies that and . From this, we conclude that, every weakly 1-absorbing prime ideal of is not weakly prime ideal of .
Theorem 45. Every weakly 1-absorbing prime ideal of is a weakly 2-absorbing ideal of .
Proof. It is clear.
The following example show that every weakly 1-absorbing prime ideal of is not 1-absorbing prime ideal of .
Example 10. Let be an ADL discussed in Example 6. Let . Clearly is a weakly 1-absorbing prime ideal of . But is not 1-absorbing prime ideal of , since implies and , for all . Thus, every weakly 1-absorbing prime ideal of is not 1-absorbing prime ideal of .
Theorem 46. Let and be ADLs and be a lattice homomorphism. Then, the following holds. (1)If is a monomorphism and is a weakly 1-absorbing prime ideal of , then is a weakly 1-absorbing prime ideal of (2)If is an epimorphism and is a weakly 1-absorbing prime ideal of such that , then is a weakly 1-absorbing prime ideal of
Proof. (1)Let for some Then by assumption, , for some . Since is monomorphism, we have . Also, since is a weakly 1-absorbing prime ideal of , we conclude either or which implies that or . Thus, is a weakly 1-absorbing prime ideal of .(2)Assume that for some . Since is an epimorphism, then there exists such that and . Then, . Since , then we get that . As is a weakly 1-absorbing prime ideal of , we have either or and which implies that or . Therefore, is a weakly 1-absorbing prime ideal of
5. Conclusion
In this paper, the concepts of 2-absorbing ideal, 1-absorbing prime ideal, weakly 1-absorbing prime ideal, and weakly 2-absorbing ideal of an almost distributive lattice are introduced and obtain certain results of these.
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Conflicts of Interest
The author declares that he/she has no conflicts of interest.