# Annihilators in Universal Algebras: A New Approach

Authors: Gezahagne Mulat Addis
Journal: Journal of Mathematics
 (2020)

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## Abstract

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


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## Body

## 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 0 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 c-unit elements in an algebra A and obtain some of their properties. By applying the concept of relative annihilators, it is shown that the class of ideals of an algebra A with a c-unit forms a complete residuated lattice. We also obtain a class of algebras called c-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 0, C-algebras, Boolean rings, and regular rings. It is observed that the commutator of ideals in c-idempotent algebras is the same as their intersection. Moreover, let every singleton set a in an algebra A be an m−-system, which is a necessary and sufficient condition for A to be c-idempotent. Furthermore, c-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,A∈K, where K is a class of algebras of a fixed type Ω, and we assume that there is an equationaly definable constant in all algebras of K denoted by 0. For a positive integer n, we write a⟶ to denote the n-tuple a1,a2,…,an∈An. For the standard concepts in universal algebras, we refer to [32–34].Definition 1 (see [29]).
A termpx⟶,y⟶ is said to be an ideal term in y⟶ if(1)pa1,…,an,0,0,…,0=0,for all a1,…,an∈A. A nonempty subset I of A will be called an ideal of A if pa1,…,an,b1,…,bm∈I for all a1,…,an∈A,b1,…,bm∈I and any ideal term px⟶,y⟶ in y⟶.
We denote the class of all ideals ofA by iA. It is easy to check that the intersection of any family of ideas of A is an ideal. So, for a subset S⊆A, always there exists a smallest ideal of A containing S which we call the ideal of A generated by S and it is denoted by S. Note that x∈S if and only if x=pa1,…,an,b1,…,bm for some a1,…,an∈A, b1,…,bm∈S, where px⟶,y⟶ is an n+m-ary ideal term in y⟶. If S=a, then we write a instead of S. In this case, x∈a if and only if x=pa1,…,an,a for some a1,…,an∈A, where px⟶,y⟶ is an n+1-ary ideal term in y⟶.Definition 2 (see [30]).
A nonzero elementu in A is said to be a formal unit if A=u, i.e., A is generated by u as an ideal.Example 1.
(1) IfG is a nontrivial cyclic group generated by the element a, then a is a formal unit.
(2) IfR is a division ring, then every nonzero element in R 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 classK of algebras is called an ideal determined class if every ideal I of A∈K is the zero congruence class of a unique congruence relation denoted by Iδ. In this case, the map I↦Iδ defines an isomorphism between the lattice of ideals and congruences on A.Definition 4 (see [29, 30]).
A termtx⟶,y⟶,z⟶ is said to be a commutator term in y⟶,z⟶ if it is an ideal term in y⟶ and an ideal term in z⟶.
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 eachI,J∈IA, their commutator I,J is defined by(2)I,J=ta⟶,i⟶,j⟶:a⟶∈An,i⟶∈Imandj⟶∈Jk,tx⟶,y⟶,z⟶is a commutator term iny⟶,z⟶.
For subsetsH,G of A, H,G denotes the product H,G. In particular, for a,b∈A, a,b is denoted by a,b.
It is immediate from the definition thatI,J is an ideal of A such that I,J⊆I∩J. 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 commutatorI,J of ideals I and J is the zero congruence class of the commutator congruence Iδ,Jδ.It is also observed that, in ideal determined varieties, the commutator of ideals is distributive over arbitrary joins of ideals. Throughout this paperK 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 algebraA is called idempotent if, for all x∈A,(3)fx,x,…,x=x,holds for each of its fundamental operations f.
It is evident thatA is idempotent if and only if every singleton set x in A is a subuniverse of A.


## 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 algebraA forms a complete residuated lattice by the use of relative annihilators.Definition 7.
LetI∈iA. For any subset S of A, we define(4)S:I=a∈A:a,S⊆I.
We callS:I, the annihilator of S relative to I.
In the following theorem, we give another description for the relative annihilator.Theorem 2.
For anyS⊆A and an ideal I of A,(5)S:I=∪H∈iA:H,S⊆I.Proof.
Let us define two setsE and G as follows:(6)E=∪H∈iA:H,S⊆I,G=a∈A:a,S⊆I.
We show thatE=G. Let a∈E. Then, there is H∈iA such that a∈H and H,S⊆I. As a∈H, we have a⊆H. This implies that a,S⊆H,S⊆I. So, a∈G and hence E⊆G. Let a∈G. Then, a,S⊆I. If we take H=a, then H∈iA such that a∈H and H,S⊆I so that a∈E and G⊆E. Therefore, E=G.
In the next theorem, we show that the annihilator of any subsetS relative to any ideal I forms an ideal by applying the above theorem.Theorem 3.
For anyS⊆A and an ideal I of A, S:I is an ideal of A.Proof.
Clearly,0,S⊆0⊆I so that 0∈S:I and hence it is nonempty. Let b⟶=b1,b2,…,bm be an m-tuples of elements from S:I. Then, by Theorem 2, there exist H1,H2,…,Hm∈iA such that each bi∈Hi and Hi,S⊆I for all i. If we put H=∨i=1mHi, then H∈iA such that each bi∈H and px⟶,y⟶. If a⟶∈An and px⟶,y⟶ is an ideal term in y⟶, then, as H is an ideal, we get pa⟶,b⟶∈H⊆S:I. Therefore, S:I is an ideal of A.Theorem 4.
For any subsetsS and T of A and ideals I and J of A, we have the following:(1)
I⊆S:I(2)
J,S⊆I if and only if J⊆S:I(3)
S⊆I⟺S:I=A(4)
I:J is the largest ideal of A such that I:J,I⊆J(5)
S⊆T⟹T:I⊆S:I(6)
I⊆J⟹S:I⊆S:J(7)
S:I∩J=S:I∩S:J(8)
I∨J:K=I:K∩J:KProof.
LetS and T be subsets of A and I,J∈iA:(1)
SinceI,S⊆I, then I⊆S:I.(2)
IfJ,S⊆I, then it is clear from the definition that J⊆S:I. Now suppose that J⊆S:I. For each a∈J, there exists Ha∈iA such that a∈Ha and Ha,S⊆I. Let us take H=∨a∈JHa, such that a∈H for all a∈J. Since the commutator , of ideals is distributive over arbitrary join, we get H,S⊆I. Now, consider(7)J,S⊆H,S⊆I.Therefore,J,S⊆I.(3)
Suppose thatS⊆I. Then, S⊆I. For any a∈A, we have a,S⊆S⊆I. Therefore, S:I=A.(4)
We first show thatI:J,I⊆J. Since I:J is an ideal of A, it can be described as follows:(8)I:J=∨a∈I:Ja.Also, we havea,I⊆J for all a∈I:J. That is,(9)∨a∈I:Ja,I⊆J.Since the commutator, of ideals is distributive over arbitrary join, it follows that I:J,I⊆J. Now, let K be any other ideal of A such that K,I⊆J. Then, it is clear from the definition that K⊆I:J. Hence, the result holds.(5)
IfS⊆T, then H,S⊆H,T for all H∈iA. This confirms that T:I⊆S:I.(6)
Suppose thatI⊆J. If H is an ideal of A such that H,S⊆I, then H,S⊆J. Therefore, S:I⊆S:J.(7)
It follows from6 that S:I∩J⊆S:I∩S:J. If a∈S:I and a∈S:J, then a,S⊆I and a,S⊆J, that is, a,S⊆I∩J so that a∈S:I∩J. Therefore, S:I∩S:J⊆S:I∩J and the equality holds.(8)
It follows from5 that I∨J:K⊆I:K∩J:K. If a∈I:K and a∈J:K, then a,I⊆K and a,J⊆K so that a,I∨J=a,I∨a,J⊆K. Therefore, a∈I∨J:K and hence the equality holds.Theorem 5.
IfJαα∈Δ is a family of ideals of A and I an ideal of A, then(10)∪α∈ΔJα:I=∩α∈ΔJα:I.Proof.
The proof follows from8 of Theorem 4, by applying the fact that the commutator , of ideals is distributive over arbitrary join.Definition 8.
A nonzero elementu∈A is a called a commutator unit (or c-unit for short) in A if a,u=a for all a∈A.Example 2.
(1) In a ringR with unity, the elements 1 and −1 are c-unit
(2) In a bounded distributive latticeL, the largest element 1 is c-unit
(3) In a bounded Hilbert algebraA, the least element 0 is c-unit
(4) Every maximal element in an almost distributive lattice isc-unit
Note also that everyc-unit element in A is a formal unit. This can be verified as follows: suppose that u∈A is c-unit. Then, for each a∈A,(11)a∈a=a,u⊆A,u⊆u.
So thatA=u and hence u is a formal unit. However, the converse does not holds in general. This can be verified in the following example.Example 3.
LetG be a nontrivial cyclic group generated by the element a. Then, a is a formal unit but not a c-unit.Remark 2.
(1)
As can be seen from1 of Example 2, c-unit (if it exists) need not necessarily be unique(2)
IfA has a c-unit u, then for each ideal I of A, it holds that I,A=I
Let us now recall the definition of residuated lattices.Definition 9.
An algebraL=L,∧,∨,⊗,⟶,0,1 of type 2,2,2,2,0,0 is called a complete residuated lattice; the following conditions are satisfied:(1)
L,∧,∨,0,1 is a complete lattice with the least element 0 and the greatest element 1.(2)
L,⊗,1 is a commutative monoid, i.e., ⊗ is commutative and associative and a⊗1=a for each a∈L.(3)
⊗ and ⟶ satisfy the adjointness property, i.e.,(12)a⊗b≤c,if and only ifa≤b⟶c,for eacha,b,c∈L (≤ denotes the lattice ordering).Theorem 6.
Suppose thatc has c-unit elements. If the commutator , of ideals is associative, then iA,∩,∨,⊗,⟶,0,A is a complete residuated lattice, where the binary operations ⊗ and ⟶ on iA are defined as follows. For each I,J∈iA,(13)I⊗J=I,J,I⟶J=I:J.Proof.
It is observed in [29] that the lattice iA,∩,∨,0,A is a complete lattice withe least element 0 and the largest element A. It is also clear from our assumption that iA,⊗,A is a commutative monoid. Moreover, it is observed in Theorem 4 (2) that ⊗ and ⟶ satisfy adjointness property. Therefore, iA,∩,∨,⊗,⟶,0,A is a complete residuated lattice.Corollary 1.
If the commutator of ideals inA coincides with their intersection, then iA forms a Heyting algebra, where ⟶ is as given in the previous theorem.Distributive lattices with least element0, c-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 elementa in a ring R is said to be idempotent if a2=a. By imitating this property to the general case of universal algebras, we define the following.Definition 10.
An elementa∈A is called idempotent if(14)a,a=a.
If all elements ofA are idempotent, then we call A a commutator idempotent algebra (or a c-idempotent algebra for short). In other words, the commutator of ideals in A is idempotent.
Boolean rings and, more generally, lattices with least element0 are the most natural examples of c-idempotent algebras. One can also verify that regular rings (a ring R is regular if for each x∈R there exists a∈R such that xax=x) are examples of c-idempotent algebras.Remark 3.
The prefix “c-” standing for the word “commutator” is added to differentiate with those idempotent algebras defined as in Definition 6.
One of the most important property ofc-idempotent algebras is that the commutator of ideals coincides with their intersection. The following definition is taken from [30].Definition 11.
A nonempty subsetM of A is said to be an m-system if M∩a,b≠∅ for all a,b∈M.
The following theorem gives a necessary and sufficient condition for an algebraA to be c-idempotent.Theorem 7.
An algebraA is c-idempotent if and only if every singleton set a in A is an m-system.Theorem 8.
(the separation axiom). LetA be a c-idempotent algebra, J∈iA and a∈A such that a∉J. Then, J and a can be separated by a prime ideal, i.e., there is a prime ideal P of A containing J and not containing a.Proof.
Let us define a set:(15)F=H∈iA:J⊆H,a∉H.
Then,F is nonempty, and it forms a poset under the usual inclusion order. Moreover, one can easily verify that F satisfies the hypothesis of Zorn’s lemma so that we can choose a maximal element, say M in F. That is, M∈iA such that J⊆M and a∉M. Now, it is enough to show that M is prime, suppose not. Then, there exist x,y∈A such that x,y⊆M but x,y∉M. Put M1=M∨x and M2=M∨y. Then, M1 and M2 are ideals of A properly containing M. By the maximality of M in F, both M1 and M2 do not belong to F. So, a∈M1 and a∈M2 which gives a,a⊆M1,M2⊆M. Since A is a c-idempotent algebra, we get a∈M, which is a contradiction. Therefore, M is a prime ideal of A such that J⊆M and a∉M.Theorem 9.
For eachI,J∈iA,(16)I:J⊆∩P:Pis a prime ideal ofA,J⊆PandI⊈P.
Moreover, ifA is c-idempotent, then the equality holds.Proof.
LetP be any prime ideal of A such that J⊆P and I⊈P. If a∈I:J, then a,I⊆J⊆P. So, a,I⊆P. Since P is prime and I⊈P, we get a∈P. Therefore,(17)I:J⊆∩P:Pis a prime ideal ofA,J⊆PandI⊈P.
To prove the other inclusion, assume thatA is c-idempotent and let a∉I:J. Then, a,I⊈J. Let us choose b∈a,I such that b∉J. By the separation axiom, there exists a prime ideal P of A such that J⊆P and b∉P which gives a,I⊈P. That is, a∉P and I⊈P. Hence,(18)∩P:Pis a prime ideal ofA,J⊆PandI⊈P⊆I:J,and therefore the equality holds.Corollary 2.
For every idealJ of a c-idempotent algebra A, the following holds:(19)J=∩P:Pis a prime ideal ofAsuch thatJ⊆P.Proof.
We first show thatA:J=J. Clearly, J⊆A:J. Let x∈A:J. Then, x,A⊆J. Since A is a c-idempotent algebra, it follows that x∈x=x,x⊆x,A⊆J 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 ofA.Definition 12.
For a subsetS of A, the annihilator of S denoted by S∗ is defined to be(20)S∗=S:0.
IfS=a, then we denote S∗ by a∗.
It is proved in the previous section thatS∗ is an ideal of A and S∗=S∗ for all S⊆A. Also, it can be verified that 0∗=A and if A has a c-unit or A is c-idempotent, then A∗=0. The following corollary gives a prime representation for annihilator of a set in c-idempotent algebras.Corollary 3.
For any subsetS of a c-idempotent algebra A,(21)S∗=∩P:Pis a prime ideal ofAsuch thatS⊈P.Proof.
The proof follows from Theorem9.Lemma 1.
For any subsetsS and T of A,(1)
S∗=∩a∈Sa∗(2)
S⊆S∗∗(3)
S∗∗∗=S∗(4)
S⊆T⟹T∗⊆S∗(5)
S⊆T⟹S∗∗⊆T∗∗(6)
S∪T∗=S∗∩T∗Theorem 10.
The following holds for allI,J∈iA:(1)
I,I∗=0, and if A is c-idempotent, then I∩I∗=0(2)
I,J=0 if and only if I⊆J∗(3)
I∨J∗=I∗∩J∗Theorem 11.
LetA be a c-idempotent algebra. Then,(22)I∩J∗∗=I∗∗∩J∗∗.Proof.
It is clear thatI∩J∗∗⊆I∗∗∩J∗∗, and we proceed to show the other inclusion. Let a∈I∗∗ and a∈J∗∗. Then, a,I∗=0 and b,J∗=0. That is, if b∈A such that either b,I=0 or b,J=0, then a,b=0. Now, for any b∈A, consider the following:(23)b∈I∩J∗⟹b,I∩J=0⟹b,I,J=0⟹b,I⊆J∗⟹a,b,I=0⟹a,b⊆I∗⟹a,a,b=0⟹a,a,b=0⟹a,b=0⟹a∈I∩J∗∗.
Therefore,I∗∗∩J∗∗⊆I∩J∗∗ and the equality holds.Definition 13.
An idealI of A is called annihilator ideal if I=S∗ for some nonempty subset S of A.
It is clear to see that0 is an annihilator ideal. Moreover, if either A has a c-unit or A is c-idempotent, then it is an annihilator ideal. We denote by i∗A the class of annihilator ideals of A.Lemma 2.
An idealI of A is an annihilator ideal if and only if I∗∗=I.
Let us consider a mapϕ:iA⟶iA defined by(24)ϕI=I∗∗,for all I∈iA. Then, the results in Lemma 1 confirm that ϕ is a closure operator on iA and the closed elements with respect to ϕ are those annihilator ideals of A.Theorem 12.
LetI,J∈iA. Then, I⊔J=I∗∩J∗∗ is the smallest annihilator ideal of A containing both I and J.Proof.
Clearly,I⊔J is an annihilator ideal of A. Also, since I∗∩J∗⊆I∗, we get I⊆I∗∗⊆I∗∩J∗∗=I⊔J. Similarly, J⊆J∗∗⊆I∗∩J∗∗=I⊔J. Now, let K be any other annihilator ideal containing both I and J. Then, K∗⊆I∗∩J∗. So, I⊔J=I∗∩J∗∗⊆K∗∗=K and this completes the proof.Theorem 13.
LetA be a c-idempotent algebra. Then, i∗A,∩,⊔,∗,0,A is a Boolean algebra.Proof.
It follows from Theorems11 and 12 that i∗A is closed under ∩ and ⊔, respectively. Since A is c-idempotent, the commutator of ideals coincides with their intersection. So, ∩ is distributive over the join ∨ of ideals. For any I,J,K∈i∗A, consider the following:(25)I∩J⊔K=I∩J∗∩K∗∗=I∩J∨K∗∗=I∗∗∩J∨K∗∗=I∩J∨K∗∗=I∩J∨I∩K∗∗=I∩J∗∩I∩K∗∗=I∩J⊔I∩K.
Therefore,i∗A,∩,⊔,0,A is a bounded distributive lattice. Moreover, for each I∈i∗A, I∩I∗=0 and I⊔I∗=I∗∩I∗∗∗=I∗∩I∗=0∗=A. That is, every element of i∗A is complemented and hence it is a Boolean algebra.Definition 14.
An idealI of A is called a dense ideal if I∗=0; otherwise, I is called a nondense ideal.
IfA has a c-unit element, then it is a dense ideal. Moreover, every maximal ideal in A is either dense or an annihilator ideal.Theorem 14.
If the commutator, of ideals is associative, then the class D of all dense ideals of A is either empty or a dual-ideal (filter) of the lattice iA.Proof.
Suppose thatD is nonempty. Let I,J∈D. Then, I∗=0=J∗. For any x∈A, consider the following:(26)x∈I∩J∗⟹x,I∩J=0⟹x,I,J=0sinceI,J⊆I∩J⟹x,I,J=0⟹x,I⊆J∗=0⟹x,I=0⟹x∈I∗=0⟹x=0.
Thus,I∩J∈D. Also, let I∈D and J∈iA. Then, I∗=0 and hence we have I∨J∗=I∗∩J∗=0. Thus, I∨J∈D, and this completes the proof.Lemma 3.
LetA be a c-idempotent algebra and I∈iA. Then,(27)I∗⊆I⟹I∗=0.Proof.
Suppose thatI∗⊆I and let a∈I∗. Then, a∈I and a,a⊆I∗,I=0. Since a∈a,a, it holds a=0. Thus, I∗=0.Theorem 15.
LetA be a c-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 forma∗(3)
For allI∈iA, I∗∨I∗∗=A(4)
For allI,J∈iA, I∩J=0 implies I∗∨J∗=AProof.
LetP be a prime ideal of A.(1)
We show thatP=P∗∗. It is enough to show that P∗∗⊆P. Let a∈P∗∗. Then, a,P∗=0⊆P. Since P is prime, either a∈P or P∗⊆P. Since every prime ideal is nondense, it follows from Lemma 3 that P∗⊆P is impossible. So, a∈P and hence the equality holds. Therefore, P is an annihilator ideal.(2)
LetP be a prime ideal of A. Then, by our assumption, P is nondense. There is a nonzero a∈A such that a,P=0. Our claim is to show that P=a∗. By (1), we have P=P∗∗, and since a∈P∗, we get P=P∗∗⊆a∗. To prove the other inclusion, let b∈a∗. Then, a,b=0⊆P. Since P is prime, either a∈P or b∈P. If a∈P, then we have the following:(28)a∈a,a⊆P∗,P=0,which is a contradiction. Therefore,a∉P and hence b∈P. So, a∗⊆P and the equality holds.(3)
LetI∈iA. Suppose on contrary I∗∨I∗∗≠A. Then, by applying Zorn’s lemma we can find a prime ideal P of A such that I∗∨I∗∗⊆P. So, I∗⊆P and I∗∗⊆P, which gives P∗⊆I∗. That is, P∗⊆P. By Lemma 3, we get P∗=0 and hence P is dense, which is a contradiction. Therefore, I∗∨I∗∗=A.(4)
LetI,J∈iA such that I,J=0. Then, J⊆I∗. So, I∗∗⊆J∗, which gives I∗∨I∗∗⊆I∗∨J∗. It follows from 3 that I∗∨J∗=A.Theorem 16.
LetA be a c-idempotent algebra. Then, the following conditions are equivalent:(1)
Every proper ideal inA is nondense(2)
Every prime ideal inA is nondense(3)
Every prime ideal is an annihilator ideal(4)
Every ideal is an annihilator idealProof.
1⟹2 is straight forward and 2⟹3 is proved in the above theorem. We proceed to show 3⟹4. Assume 3. Let I be any ideal in A. We need to show that I∗∗⊆I, suppose not. Then, there is x∈I∗∗ such that x∉I. By the separation axiom, there is a prime ideal P of A such that I⊆P and x∉P. This gives I∗∗⊆P∗∗. It follows from our assumption that I∗∗⊆P, which is a contradiction. Therefore, I∗∗⊆I and hence it is an annihilator ideal. 4⟹1 is also straight forward.Theorem 17.
LetA be a c-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 idealProof.
1⟹2. Assume 1 and let P be a prime ideal of A. Then, P=P∗∗. Let Q be any prime ideal of A such that Q⊆P. Our aim is to show that Q=P, suppose not. Then, there is x∈P−Q. For any a∈P∗, a,x=0⊆Q. Since Q is prime and x∉Q, it yields that a∈Q⊆P. Hence, a∈P∩P∗=0 so that P=P∗∗=A which is a contradiction.2⟹3. Assume 2. Let P be a prime ideal of A and J a proper ideal of A such that P⊆J. By the separation axiom, there exists a prime ideal Q of A such that J⊆Q so that P⊆P. By P⊆P, Q is minimal and hence P=Q. Therefore, P=J and hence P is maximal.3⟹1. Assume 3 and let P be any prime ideal. Then, it is maximal. It is also clear that P⊆P∗∗ and by our assumption P∗∗ is proper. Since P is maximal it yields P=P∗∗ 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.
LetK be a class of algebras of the same type Ω with 0. K is said to be an annihilator class, whenever it has a so-called annihilator term, i.e., a binary term qx,y of type Ω satisfying the following conditions:(A1)qx,0=0=q0,x(A2) For eachA∈K, each ideal term tx⟶,y⟶ in y⟶ and every a1,…,an,b1,…,bm,c∈A, qbi,c=0 for all i=1,2,…,m implies qta1,…,an,b1,…,bm,c=0.(A3) For eachA∈K and each b∈A, b=qx,b:x∈A.Definition 16.
LetK be an annihilator class and qx,y its annihilator term. Let A∈K, b∈A and ∅≠M⊆A. The set,(29)Ab=y∈A:qy,b=0,will be called the annihilator of the element b in the sense of Chajda and Halas. The set,(30)AM=∩Ab:b∈M,will be called the annihilator of M in the sense of Chajda and Halas.Definition 17.
A classK of algebras is said to satisfy the property M if for each A∈K and each a,b∈A; there exists a unique element in A denoted by pa,b such that a,b=pa,b. We call pa,b the product of a and b.
The variety of groups satisfies the propertyM, where px,y=x−1y−1xy. The variety of rings has the property M, where px,y=xy+yx. In particular, the class of commutative rings has this property with the product px,y=xy. Moreover, the binary term px,y=x∧y is a product term in the class of distributive lattices.Theorem 18.
IfK is a class of algebras satisfying the property M, then it has an annihilator term.Proof.
It is enough to show thatpx,y is the annihilator term.(1)
Sincex,0=0 for all x∈A, it follows that(31)px,0=0=p0,x.(2)
Lettx⟶,y⟶ be any ideal term in y⟶, a1,…,an,b1,…,bm,c∈A such that pbi,c=0 for all i=1,2,…,m, i.e., bi,c=0 for each i, which implies that(32)∨i=1mbi,c=b1,…,bm,c=0,which gives thatx,c=0 for all x∈b1,…,bm. Being tx⟶,y⟶ an ideal term in y⟶, we have ta1,…,an,b1,…,bm∈b1,…,bm and hence(33)ta1,…,an,b1,…,bm,c=0,means thatpta1,…,an,b1,…,bm,c=0.(3)
Leta∈A and H=px,a:x∈A. Our aim is to show that H=a. If z∈H, then z=pb,a for some b∈A, i.e., z=a,b. Since a,b⊆a, it holds that z∈a and hence H⊆a. To prove the other inclusion, let x∈a. Then, there exist an n+1-ary ideal term tx⟶,y in y, a1,…,an∈A such that(34)x=ta1,…,an,a.
Ifx=0, then it belongs to H. Assume that x≠0. Now, let us define an n+1+1-ary term qx⟶,y,z as follows:(35)qx1,…,xn,y,z=0,ifz=0,tx1,…,xn,y,botherwise.
Then,qx⟶,y,z is a commutator term in y and z such that qa1,…,an,a,x=x which implies that x=a,x, i.e., pa,x=x, so that x∈H and hence a⊆H. Therefore, the equality holds and this completes the proof.
The above theorem shows that, for any subsetS of A in a class of algebras satisfying the property M, the annihilator S∗ of S coincides with that of the annihilator AS of S in the sense of Chajda and Halas.



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