#### Abstract

The purpose of this paper is to study annihilators and annihilator ideals in a more general context; in universal algebras.

#### 1. Introduction

Annihilators and annihilator ideals have been studied in different classes of algebras. In ring theory, Bear rings, quasi-Bear rings, and principally quasi-Baer rings are defined using left and right annihilator ideals (see [1–6]). The notion of annihilators is also closely connected to that of regular rings, specifically to biregular rings, Von Neumann regular rings, self-injective rings, and continuous regular rings (see [7–12]). More recently, annihilators were studied in local cohomology modules (see [13–16]).

On the contrary, the concept of annihilators in lattices was studied by Mandelker [17] and later extended to the class of distributive lattices by Cornish [18], Speed [19], and Davey [20]. In [21], Davey and Nieminen studied the structure of annihilators in modular lattices. Halaš [22] studied annihilators in ordered sets as a generalization of lattices. There are also other classes of algebras for which annihilators and annihilator ideals have a great deal of applications. For instance, a class of BCK-algebras [23], Banach algebras [24], standard QBCC algebras [25], weakly standard BCC-algebras [26], C-algebras [27, 28], and others.

In recent times, the theory of ideals has been taking place in a more general context. Gumm and Ursini [29] introduced the concept of ideals in universal algebras having a constant as a generalization of those familiar structures: normal subgroups (in groups), ideals (in rings), submodules (in modules), subspaces (in vector spaces), and filters (in implication algebras or Heyting algebras). They have also defined and characterized the commutator (or the product) of ideals in universal algebras. Later, Ursini [30] applied this product to define and study prime ideals in universal algebras. He has also defined relative annihilator of ideals under a name “residual of ideals.”

The general concept of annihilators in universal algebras was introduced by Chajda and Halas in the paper [31] based on annihilator terms. In the present paper, we propose another approach to study annihilators and annihilator ideals in universal algebras by the use of the commutator of ideals in the sense of Gumm and Ursini [29]. We define -unit elements in an algebra and obtain some of their properties. By applying the concept of relative annihilators, it is shown that the class of ideals of an algebra with a -unit forms a complete residuated lattice. We also obtain a class of algebras called -idempotent algebras, for which the lattice of all annihilator ideals of each algebra forms a Boolean algebra. This class of algebras contains well-known structures, lattices with least element , -algebras, Boolean rings, and regular rings. It is observed that the commutator of ideals in -idempotent algebras is the same as their intersection. Moreover, let every singleton set in an algebra be an -system, which is a necessary and sufficient condition for to be -idempotent. Furthermore, -idempotent algebras satisfy an important property, known as the separation axiom.

#### 2. Preliminaries

This section contains some definitions and results which will be used in this paper. Throughout this paper, , where is a class of algebras of a fixed type , and we assume that there is an equationaly definable constant in all algebras of denoted by . For a positive integer , we write to denote the -tuple . For the standard concepts in universal algebras, we refer to [32–34].

*Definition 1 (see [29]). *A term is said to be an ideal term in iffor all . A nonempty subset of will be called an ideal of if for all and any ideal term in .

We denote the class of all ideals of by . It is easy to check that the intersection of any family of ideas of is an ideal. So, for a subset , always there exists a smallest ideal of containing which we call the ideal of generated by and it is denoted by . Note that if and only if for some , , where is an -ary ideal term in . If , then we write instead of . In this case, if and only if for some , where is an -ary ideal term in .

*Definition 2 (see [30]). *A nonzero element in is said to be a formal unit if , i.e., is generated by as an ideal.

*Example 1. *(1) If is a nontrivial cyclic group generated by the element , then is a formal unit.

(2) If is a division ring, then every nonzero element in is a formal unit.

(3) The largest element in a bounded distributive lattice is a formal unit.

(4) Every maximal element in an almost distributive lattice is a formal unit.

*Remark 1. *A formal unit in an algebra (if it exists) is not necessarily unique (e.g., cyclic groups and almost distributive lattices may have several formal units).

*Definition 3 (see [29]). *A class of algebras is called an ideal determined class if every ideal of is the zero congruence class of a unique congruence relation denoted by . In this case, the map defines an isomorphism between the lattice of ideals and congruences on .

*Definition 4 (see [29, 30]). *A term is said to be a commutator term in if it is an ideal term in and an ideal term in .

An operation called the commutator (or the product) of ideals proposed by Gumm and Ursini [29] and later studied by Ursini [30] will be considered useful for our purposes.

*Definition 5 (see [30]). *For each , their commutator is defined byFor subsets of , denotes the product . In particular, for , is denoted by .

It is immediate from the definition that is an ideal of such that . Moreover, one can observe that is commutative and increasing in both arguments. The following theorem gives a characterization for the commutator of ideals using the general commutator of congruences in ideal determined varieties. About the commutator of congruences see [35, 36].

Theorem 1 (see [30]). *In an ideal determined variety, the commutator of ideals and is the zero congruence class of the commutator congruence .*

It is also observed that, in ideal determined varieties, the commutator of ideals is distributive over arbitrary joins of ideals. Throughout this paper is assumed to be an ideal determined variety. Further detail on ideals (and their commutator) in universal algebras can be found in [30, 37–42].

*Definition 6 (see [43]). *An algebra is called idempotent if, for all ,holds for each of its fundamental operations .

It is evident that is idempotent if and only if every singleton set in is a subuniverse of .

#### 3. Relative Annihilators

In this section, we define relative annihilators in universal algebras and prove some of their basic properties. Mainly, we show that the class of ideals in an algebra forms a complete residuated lattice by the use of relative annihilators.

*Definition 7. *Let . For any subset of , we defineWe call , the annihilator of relative to .

In the following theorem, we give another description for the relative annihilator.

Theorem 2. *For any and an ideal of ,*

*Proof. *Let us define two sets and as follows:We show that . Let . Then, there is such that and . As , we have . This implies that . So, and hence . Let . Then, . If we take , then such that and so that and . Therefore, .

In the next theorem, we show that the annihilator of any subset relative to any ideal forms an ideal by applying the above theorem.

Theorem 3. *For any and an ideal of , is an ideal of .*

*Proof. *Clearly, so that and hence it is nonempty. Let be an -tuples of elements from . Then, by Theorem 2, there exist such that each and for all . If we put , then such that each and . If and is an ideal term in , then, as is an ideal, we get . Therefore, is an ideal of .

Theorem 4. *For any subsets and of and ideals and of , we have the following:*(1)*(2)** if and only if *(3)*(4)** is the largest ideal of such that *(5)*(6)**(7)**(8)*

*Proof. *Let and be subsets of and :(1)Since , then .(2)If , then it is clear from the definition that . Now suppose that . For each , there exists such that and . Let us take , such that for all . Since the commutator of ideals is distributive over arbitrary join, we get . Now, consider Therefore, .(3)Suppose that . Then, . For any , we have . Therefore, .(4)We first show that . Since is an ideal of , it can be described as follows: Also, we have for all . That is, Since the commutator of ideals is distributive over arbitrary join, it follows that . Now, let be any other ideal of such that . Then, it is clear from the definition that . Hence, the result holds.(5)If , then for all . This confirms that .(6)Suppose that . If is an ideal of such that , then . Therefore, .(7)It follows from that . If and , then and , that is, so that . Therefore, and the equality holds.(8)It follows from that . If and , then and so that . Therefore, and hence the equality holds.

Theorem 5. *If is a family of ideals of and an ideal of , then*

*Proof. *The proof follows from of Theorem 4, by applying the fact that the commutator of ideals is distributive over arbitrary join.

*Definition 8. *A nonzero element is a called a commutator unit (or -unit for short) in if for all .

*Example 2. *(1) In a ring with unity, the elements and are -unit

(2) In a bounded distributive lattice , the largest element is -unit

(3) In a bounded Hilbert algebra , the least element is -unit

(4) Every maximal element in an almost distributive lattice is -unit

Note also that every -unit element in is a formal unit. This can be verified as follows: suppose that is -unit. Then, for each ,So that and hence is a formal unit. However, the converse does not holds in general. This can be verified in the following example.

*Example 3. *Let be a nontrivial cyclic group generated by the element . Then, is a formal unit but not a -unit.

*Remark 2. * (1)As can be seen from of Example 2, -unit (if it exists) need not necessarily be unique(2)If has a -unit , then for each ideal of , it holds that Let us now recall the definition of residuated lattices.

*Definition 9. *An algebra of type is called a complete residuated lattice; the following conditions are satisfied:(1) is a complete lattice with the least element and the greatest element .(2) is a commutative monoid, i.e., is commutative and associative and for each .(3) and satisfy the adjointness property, i.e., for each ( denotes the lattice ordering).

Theorem 6. *Suppose that has -unit elements. If the commutator of ideals is associative, then is a complete residuated lattice, where the binary operations and on are defined as follows. For each ,*

*Proof. *It is observed in [29] that the lattice is a complete lattice withe least element and the largest element . It is also clear from our assumption that is a commutative monoid. Moreover, it is observed in Theorem 4 (2) that and satisfy adjointness property. Therefore, is a complete residuated lattice.

Corollary 1. *If the commutator of ideals in coincides with their intersection, then forms a Heyting algebra, where is as given in the previous theorem.*

Distributive lattices with least element , -algebras, and almost distributive lattices are examples of algebras in which the commutator is the same as the set theoretic intersection, and hence their lattice of ideals forms a Heyting algebra.

An element in a ring is said to be idempotent if . By imitating this property to the general case of universal algebras, we define the following.

*Definition 10. *An element is called idempotent ifIf all elements of are idempotent, then we call a commutator idempotent algebra (or a -idempotent algebra for short). In other words, the commutator of ideals in is idempotent.

Boolean rings and, more generally, lattices with least element are the most natural examples of -idempotent algebras. One can also verify that regular rings (a ring is regular if for each there exists such that ) are examples of -idempotent algebras.

*Remark 3. *The prefix “-” standing for the word “commutator” is added to differentiate with those idempotent algebras defined as in Definition 6.

One of the most important property of -idempotent algebras is that the commutator of ideals coincides with their intersection. The following definition is taken from [30].

*Definition 11. *A nonempty subset of is said to be an -system if for all .

The following theorem gives a necessary and sufficient condition for an algebra to be -idempotent.

Theorem 7. *An algebra is -idempotent if and only if every singleton set in is an -system.*

Theorem 8. *(the separation axiom). Let be a -idempotent algebra, and such that . Then, and can be separated by a prime ideal, i.e., there is a prime ideal of containing and not containing .*

*Proof. *Let us define a set:Then, is nonempty, and it forms a poset under the usual inclusion order. Moreover, one can easily verify that satisfies the hypothesis of Zorn’s lemma so that we can choose a maximal element, say in . That is, such that and . Now, it is enough to show that is prime, suppose not. Then, there exist such that but . Put and . Then, and are ideals of properly containing . By the maximality of in , both and do not belong to . So, and which gives . Since is a -idempotent algebra, we get , which is a contradiction. Therefore, is a prime ideal of such that and .

Theorem 9. *For each ,**Moreover, if is -idempotent, then the equality holds.*

*Proof. *Let be any prime ideal of such that and . If , then . So, . Since is prime and , we get . Therefore,To prove the other inclusion, assume that is -idempotent and let . Then, . Let us choose such that . By the separation axiom, there exists a prime ideal of such that and which gives . That is, and . Hence,and therefore the equality holds.

Corollary 2. *For every ideal of a -idempotent algebra , the following holds:*

*Proof. *We first show that . Clearly, . Let . Then, . Since A is a -idempotent algebra, it follows that and hence the equality holds. Thus, the result follows from Theorem 9.

#### 4. Annihilator Ideals

In this section, we study annihilator ideals in universal algebras. We begin by defining the annihilator of subsets of .

*Definition 12. *For a subset of , the annihilator of denoted by is defined to beIf , then we denote by .

It is proved in the previous section that is an ideal of and for all . Also, it can be verified that and if has a -unit or is c-idempotent, then . The following corollary gives a prime representation for annihilator of a set in -idempotent algebras.

Corollary 3. *For any subset of a -idempotent algebra ,*

*Proof. *The proof follows from Theorem 9.

Lemma 1. *For any subsets and of ,*(1)*(2)**(3)**(4)**(5)**(6)*

Theorem 10. *The following holds for all :*(1)*, and if is -idempotent, then *(2)* if and only if *(3)

Theorem 11. *Let be a -idempotent algebra. Then,*

*Proof. *It is clear that , and we proceed to show the other inclusion. Let and . Then, and . That is, if such that either or , then . Now, for any , consider the following:Therefore, and the equality holds.

*Definition 13. *An ideal of is called annihilator ideal if for some nonempty subset of .

It is clear to see that is an annihilator ideal. Moreover, if either has a -unit or is -idempotent, then it is an annihilator ideal. We denote by the class of annihilator ideals of .

Lemma 2. *An ideal of is an annihilator ideal if and only if .**Let us consider a map defined byfor all . Then, the results in Lemma 1 confirm that is a closure operator on and the closed elements with respect to are those annihilator ideals of .*

Theorem 12. *Let . Then, is the smallest annihilator ideal of containing both and .*

*Proof. *Clearly, is an annihilator ideal of . Also, since , we get . Similarly, . Now, let be any other annihilator ideal containing both and . Then, . So, and this completes the proof.

Theorem 13. *Let be a -idempotent algebra. Then, is a Boolean algebra.*

*Proof. *It follows from Theorems 11 and 12 that is closed under and , respectively. Since is -idempotent, the commutator of ideals coincides with their intersection. So, is distributive over the join of ideals. For any , consider the following:Therefore, is a bounded distributive lattice. Moreover, for each , and . That is, every element of is complemented and hence it is a Boolean algebra.

*Definition 14. *An ideal of is called a dense ideal if ; otherwise, is called a nondense ideal.

If has a -unit element, then it is a dense ideal. Moreover, every maximal ideal in is either dense or an annihilator ideal.

Theorem 14. *If the commutator of ideals is associative, then the class of all dense ideals of is either empty or a dual-ideal (filter) of the lattice .*

*Proof. *Suppose that is nonempty. Let . Then, . For any , consider the following:Thus, . Also, let and . Then, and hence we have . Thus, , and this completes the proof.

Lemma 3. *Let be a -idempotent algebra and . Then,*

*Proof. *Suppose that and let . Then, and . Since , it holds . Thus, .

Theorem 15. *Let be a -idempotent algebra in which every prime ideal is nondense. Then, the following conditions hold:*(1)*Every prime ideal is an annihilator ideal*(2)*Every prime ideal is of the form *(3)*For all , *(4)*For all , implies *

*Proof. *Let be a prime ideal of .(1)We show that . It is enough to show that . Let . Then, . Since is prime, either or . Since every prime ideal is nondense, it follows from Lemma 3 that is impossible. So, and hence the equality holds. Therefore, is an annihilator ideal.(2)Let be a prime ideal of . Then, by our assumption, is nondense. There is a nonzero such that . Our claim is to show that . By (1), we have , and since , we get . To prove the other inclusion, let . Then, . Since is prime, either or . If , then we have the following: which is a contradiction. Therefore, and hence . So, and the equality holds.(3)Let . Suppose on contrary . Then, by applying Zorn’s lemma we can find a prime ideal of such that . So, and , which gives . That is, . By Lemma 3, we get and hence is dense, which is a contradiction. Therefore, .(4)Let such that . Then, . So, , which gives . It follows from that .

Theorem 16. *Let be a -idempotent algebra. Then, the following conditions are equivalent:*(1)*Every proper ideal in is nondense*(2)*Every prime ideal in is nondense*(3)*Every prime ideal is an annihilator ideal*(4)*Every ideal is an annihilator ideal*

*Proof. * is straight forward and is proved in the above theorem. We proceed to show . Assume . Let be any ideal in . We need to show that , suppose not. Then, there is such that . By the separation axiom, there is a prime ideal of such that and . This gives . It follows from our assumption that , which is a contradiction. Therefore, and hence it is an annihilator ideal. is also straight forward.

Theorem 17. *Let be a -idempotent algebra. If every prime ideal is nondense, then the following conditions are equivalent:*(1)*Every prime ideal is an annihilator ideal*(2)*Every prime ideal is a minimal prime ideal*(3)*Every prime ideal is a maximal ideal*

*Proof. *. Assume and let be a prime ideal of . Then, . Let be any prime ideal of such that . Our aim is to show that , suppose not. Then, there is . For any , . Since is prime and , it yields that . Hence, so that which is a contradiction.

. Assume . Let be a prime ideal of and a proper ideal of such that . By the separation axiom, there exists a prime ideal of such that so that . By , is minimal and hence . Therefore, and hence is maximal.

. Assume and let be any prime ideal. Then, it is maximal. It is also clear that and by our assumption is proper. Since is maximal it yields and hence it is an annihilator ideal. The proof ends.

#### 5. Annihilators in the Sense of Chajda and Halas

Annihilators in universal algebras have been first studied by Chajda and Halas in the paper [31]. They have used a different approach by introducing the notion of annihilator terms. In this section, we obtain a class of algebras for which the present definition of annihilators is the same as that of annihilators studied by Chajda and Halas. Let us first recall the definition of annihilators proposed by Chajda and Halas in the paper [31].

*Definition 15. *Let be a class of algebras of the same type with 0. is said to be an annihilator class, whenever it has a so-called annihilator term, i.e., a binary term of type satisfying the following conditions: (A1) (A2) For each , each ideal term in and every , for all implies . (A3) For each and each , .

*Definition 16. *Let be an annihilator class and its annihilator term. Let , and . The set,will be called the annihilator of the element in the sense of Chajda and Halas. The set,will be called the annihilator of in the sense of Chajda and Halas.

*Definition 17. *A class of algebras is said to satisfy the property if for each and each ; there exists a unique element in denoted by such that . We call the product of and .

The variety of groups satisfies the property , where . The variety of rings has the property , where . In particular, the class of commutative rings has this property with the product . Moreover, the binary term is a product term in the class of distributive lattices.

Theorem 18. *If is a class of algebras satisfying the property , then it has an annihilator term.*

*Proof. *It is enough to show that is the annihilator term.(1)Since for all , it follows that(2)Let be any ideal term in , such that for all , i.e., for each , which implies that which gives that for all . Being an ideal term in , we have and hence means that .(3)Let and . Our aim is to show that . If , then for some , i.e., . Since , it holds that and hence . To prove the other inclusion, let . Then, there exist an -ary ideal term in , such thatIf , then it belongs to . Assume that . Now, let us define an -ary term as follows:Then, is a commutator term in and such that which implies that , i.e., , so that and hence . Therefore, the equality holds and this completes the proof.

The above theorem shows that, for any subset of in a class of algebras satisfying the property , the annihilator of coincides with that of the annihilator of in the sense of Chajda and Halas.

#### Data Availability

No data were used to support the results of this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.