Radical theory has long played a central role in the structural investigation of associative rings, particularly in understanding how classes of rings can be characterized through hereditary, strong, and special radical properties. Foundational studies by Divinsky, Krempa, and Sulinski [1] demonstrated the importance of strong radical properties in associative rings, while the systematic exposition of radical constructions and lattice-theoretic approaches was later consolidated in [2]. Within this framework, supernilpotent radicals and their associated lattice structures have attracted considerable attention because of their close relationship with hereditary and special radicals.

One of the fundamental directions in radical theory concerns the study of atoms in lattices of radicals. The investigation of atoms of supernilpotent radicals was initiated by France-Jackson in [3], where minimal supernilpotent radicals properly containing the prime radical were characterized through the existence of ∗-rings. Further developments by Puczylowski and Roszkowska [4] clarified several structural properties of atoms in lattices of radicals, while Korolczuk [5] investigated the lattice structure of special radicals through the introduction of ∗-rings. These results established a strong connection between prime rings, special classes, and the existence of minimal radical classes.

The relationship between special radicals and prime essential rings has also become an important topic in radical theory. Gardner and Stewart [6] introduced prime essential rings and demonstrated several of their structural properties. Later, France-Jackson and Groenewald [7] showed that certain rings can generate both supernilpotent atoms and special atoms. This direction was further strengthened by Wahyuni, Wijayanti, and France-Jackson [8], who constructed a prime essential ring generating a special atom. These developments indicate that prime essential rings provide an efective mechanism for constructing minimal radicals and studying their lattice-theoretic behavior.

Parallel to these developments, the study of normal radicals and N-radicals has evolved significantly through the works of Sands [9, 10], Jaegermann [11], and Jaegermann and Sands [12]. In particular, the characterization of supernilpotent radicals that are simultaneously hereditary and strong revealed deep relationships among normal radicals, N-radicals, and A-radicals. Moreover, Beidar and Salavova [13] investigated the lattice structure of N-radicals and established several important properties of the lattice . Despite these substantial developments, the relationship between prime essential rings, supernilpotent atoms, and N-atoms remains only partially understood.

N-radicals form an important class of radicals and have been investigated by many prominent authors. In 1983, Beidar and Salavova discussed the lattice of N−radicals in [13]. Moreover, some further properties of normal radicals are explained in [11]. In fact, a supernilpotent radical that is normal forms an N−radical. The relationship between normal radicals and N−radicals are explained in [9] and [10]. On the other hand, Jaegerman and Sands also gave additional information about relationship between normal radicals, N-radicals, and A−radicals in [12].

It is well known [2] that many important radicals, such as the prime radical the locally nilpotent radical and the Jacobson radical , are N-radicals. In contrast, the Brown–McCoy radical G is not an N-radical. Moreover, the famous K¨othe problem, which asks whether the nilradical is left strong, is equivalent to the question of whether is an N-radical. Furthermore, N-radicals are linked to some other long standing open problems in radical theory, as shown in [14] and [15].

So, there is a good reason for studying N-radicals. A theoretical implementation of the Jacobson radical as N−radical can be seen in [16]. While [8] established that certain prime essential rings generate special atoms, our results explained in this paper extend this by: (1) demonstrating that some prime essential rings can generate N-atoms, supernilpotent atoms, and special atoms—and (2) revisiting the Questions 1 and 2 in [8] through the lens of N-radicals. Moreover, as one of the famous radical class, the theoretical implementation of Jacobson radical can be seen in [17].

Recent studies indicate that problems related to special radicals and supernilpotent radicals naturally extend to broader algebraic settings. For example, Prasetyo et al. [18] generalized several radical-theoretic constructions into module theory through weakly special classes of modules. Moreover, Yuwaningsih et al. [19] extended several classical concepts of module homomorphisms into the framework of (R, S)-modules, showing that radical-theoretic and module-theoretic structures continue to develop in more generalized algebraic settings. In addition, Prasetyo et al. [20] constructed supernilpotent radical classes using topological methods related to Tychonof spaces. Connections between algebraic and topological structures also appear in the work of Burgess and Raphael [21]. Furthermore, recent investigations on β-classes in [15] and the prime radical in [22] revisited several open problems related to special radicals and hereditary constructions. These developments suggest that the study of N-radicals remains highly relevant, both structurally and in terms of possible generalizations.

Motivated by the above developments, this paper investigates N-atoms as minimal elements of the lattice and studies their relationship with supernilpotent atoms in the lattice K. In particular, we revisit Questions 1 and 2 posed in [8] from the perspective of N-radicals and establish conditions under which certain prime essential rings generate N-atoms, supernilpotent atoms, and special atoms simultaneously. Hence, the present work extends several earlier results concerning special atoms and provides additional insight into the structure of lattices of N-radicals.

In this paper, all rings are associative, and all classes of rings are closed under isomorphisms and contain the one-element ring . The fundamental definitions and properties of radicals can be found in [2]. We will use the following definitions and notations. If R is a ring, then (respectively, means that I is an ideal (respectively, a left ideal, a right ideal) of R. If A is a subring of we use to denote the ideal of R generated by A. denotes the ring defined on the same underlying additive group as that of a ring R with zero multiplication [2].

A subring A of a ring R is said to be accessible (respectively, left accessible) if there exist subrings of R such that is an ideal (respectively, left ideal) of , for every $i = 1 , 2 , . . . , n - 1 [2].

A class of rings is said to be hereditary (respectively, left hereditary, right hereditary) if for every ring , we have for every ideal (respectively, left ideal, right ideal) I of . Let be a class of rings. The class is called a radical class if α satisfies the following conditions [2]:

(i) For any ring , then for every

(ii) For every ring where

(iii) for every ring A.

For any radical α and any ring denotes the α-radical of that is, the largest ideal of R that belongs to α. A hereditary class of prime rings is called a special class is closed under essential extensions [2].

The smallest radical containing a given class of rings will be denoted by . In a special case, the smallest radical containing the class of all zero rings is precisely the prime radical and it will be denoted by $\beta \ [ 2 ]$

Some results related to problems in ring radical theory have also been generalized in module theory, for example [18]. In addition, [20] explores the construction of a supernilpotent radical class using topological concept, namely a Thyconof space. The definition of a Thyconof space will be given later in the Results and Discussion Section.

A left hereditary and a left strong radical α that contains the prime radical is called an N-radical. It is well known [11] that a supernilpotent radical α is an radical if and only if α is right hereditary and right strong. Moreover, the collection of all N-radicals forms a sublattice of the lattice K of all supernilpotent radicals.

We now present the following useful previous results from [23] and [4]:

Theorem 2.1.[23] is a hereditary (respectively, left hereditary) class of rings, then the radical is hereditary (respectively, left hereditary).

Theorem 2.2.[4] Let be a homomorphically closed class of rings and let R be a semiprime ring such that . Then

(1) R contains a left accessible subring such that

(2) If the class is hereditary, then R contains a ideal S such that

We recall the following useful observation.

Definition 2.3.[2] A hereditary radical is called a supernilpotent radical. The class for every nonzero homomorphic image of A} of rings of a given special class µ forms a radical class and it is called a special radical. A radical α is called left strong (respectively, right strong), if for every ring R and every left ideal (respectively, right ideal) of we have , where is the ideal of R generated by L.

It is well known [1] that for any homomorphically closed class of rings, there exists the smallest strong radical containing

Lemma 2.4.Let be a supernilpotent radical (respectively, an N-radical). Then α is a supernilpotent atom (respectively, an N-atom) if and only if for every ring and every supernilpotent radical (respectively N-radical) , we have either

Proof. ⇒: Assume that is a supernilpotent atom and let and If α then and then . Otherwise, we have , which implies that since and α is a supernilpotent atom. But, being a supernilpotent atom, α is hereditary. So, since , it follows that which implies that . This means that 1 and, since and , it follows that . Hence,

⇐: Assume that is a supernilpotent radical such that for every ring and every supernilpotent radical , we have either . Suppose that there exists a supernilpotent radical γ such that Then, there exist rings and such that and . Then the ring and which implies that 1 and and we have a contradiction. Hence, α is a supernilpotent atom.

The same proof applies to N-radicals by replacing K with throughout. □

Let α be a supernilpotent atom (respectively, an Then Consequently, α contains the smallest supernilpotent radical (respectively, the smallest N-radical containing , for every semiprime ring . Therefore, in studying supernilpotent atoms (respectively, N-atoms), it is important to know what rings the radicals (respectively, consist of. To describe such rings, we will need the following well known fact which follows from the lower radical construction:

Proposition 3.1.[2] is a homomorphically closed class of rings, then {R : Every nonzero homomorphic image of R has a nonzero accessible subring in

We therefore have

Proposition 3.2.[2]Let be the supernilpotent radical generated by Then , where , for some accessible subring . Thus we have

Consider a class of rings. The construction of the smallest special radical containing µ has been already explained in [8]. Now, we will give an explanation of the construction of the smallest supernilpotent radical containing a given ring A. It follows from Proposition 3.2 that the semisimple class of the smallest hereditary and homomorphically closed class of rings containing A is described below

In fact, the class is essentially closed, whence the class

is essentially closed, where π is the class of all prime rings. It is clear that the class π of all prime rings is a special class since it is essentially closed and hereditary for nonzero ideals. We therefore have that the smallest special radical containing A is equal to

It was proved in [3] that, if A is a nonzero ∗-ring, that is, a semiprime ring A such that for every nonzero ideal I of $A \ [ 3 ]\bar { l } _ { A }$ is a supernilpotent atom and

Moreover, it was shown in [7] that every nonzero ∗∗-ring A also generates a supernilpotent atom , where a prime ring A is called ∗∗-ring if the smallest special class containing A is closed under semiprime homomorphic images of A.

Our next result shows how to build the smallest N-radical containing a nonzero semiprime ring R. This construction is similar to the construction of the smallest supernilpotent radical containing a nonzero semiprime ring R that was given above and it follows directly from Theorem 2.2 and Theorem 2.1 as well as from the very well known fact that the lower radical generated by a left hereditary and homomorphically closed class of rings is left hereditary.

Corollary 3.3.Let R be a nonzero semiprime ring and let denote the smallest left hereditary and homomorphically closed class of rings containing R. Then for some left accessible subring S of and

Suppose R is a nonzero semiprime ring. Then the following characterization of supernilpotent atoms(respectively, N-atoms of the form follows directly from the definition of a supernilpotent atom (respectively, an N-atom):

Lemma 3.4. Let R be a nonzero semiprime ring. The following conditions are equivalent:

(i) is a supernilpotent atom (respectively, is an N-atom).

(ii) (respectively, , for every nonzero semiprime homomorphic image R of R.

(iii) , for every nonzero semiprime ring

(iv) (respectively, , for every nonzero prime homomorphic image

(v) , for every nonzero prime ring (respectively,

Our main result describes the relationship between supernilpotent atoms and N-atoms. It also shows that N-atoms exist and how to construct some N-atoms.

Theorem 3.5.If R is a semiprime ring such that is a supernilpotent atom, then is an N-atom.

Proof. We will prove this theorem into three claims as follows.

(1) Claim I: Assume that R is a semiprime ring such that is a supernilpotent atom and let A be any nonzero semiprime ring such that . Then . We are going to prove that A contains a left ideal S such that . In view of Lemma 3.4 (iii), it is suficient to show that . Now, and the class is homomorphically closed and left hereditary (hence hereditary). So, since , it follows from Theorem 2.2 (2) that A contains a left ideal S such that that is, , for some

(2) Claim II: We will show that , where S^¯ is a homomorphic image of S and Observe that the ring A contains a left ideal such that , for some nonzero left ideal M of R. Indeed, if , then which implies that is a left ideal of R. Then and fit the bill. If , then consider . Clearly, and which implies that Moreover, since, otherwise, which gives . But this implies that is a nonzero nilpotent ring isomorphic to a nonzero semiprime ring which cannot be. Now, since and it follows that for some such that . But, since is hereditary. Hence . Then take and

Since M/I is a nonzero semiprime ring, it follows from Lemma 2.2 of [14] that there exists a semiprime homomorphic image R of R and a left ideal S of R such that so that

(3) Claim III: We will show that is an N-atom by proving is an atom of the lattice of all N−radicals. Since and radicals are closed under extensions, it follows that which implies that because and is left strong. So, because On the other hand, the semiprime ring , since is a supernilpotent atom and is a supernilpotent radical, it follows from Lemma 2.4 that and so . Moreover, since is a supernilpotent atom containing the nonzero semiprime ring , it follows from Lemma 3.4 (iii) that So, But, since N-radicals are supernilpotent, it follows that . Consequently, which gives . So, as , it follows that . Moreover, since and N-radicals are left hereditary and homomorphically closed, we have which implies that . Thus, recapping all our observations, we have , which ends the proof.

As a direct consequence of Theorem 3.5, we get the following corollary:

Corollary 3.6.If R is a nonzero ∗-ring, then is an N-atom. Furthermore, if R is a nonzero ∗∗-ring, then is also an N-atom.

Proof. Let R be a ∗−ring. Since the class ∗ of all ∗−rings is strictly contained in the class ∗∗ of all ∗ ∗ −rings [7], R is a ∗ ∗ −ring. It follows from Theorem 6 in [7] that is a supernilpotent atom. Moreover, it follows from Theorem 3.5 that is an N-atom. By using the same argument, we can directly see is an N-atom if R is a ∗ ∗ −ring. □

Theorem 3.5 also implies that some nonzero semiprime rings do not generate N-atoms, as our next results show. To identify such rings, recall that a semiprime ring R is called prime essential [6] if for every nonzero prime ideal P of R, we have whenever I is a nonzero two-sided ideal of R.

Lemma 3.7.[6] Let S be a linearly ordered set with no greatest element, with least element e and which is such that every interval has cardinality κ. Then S is a semi-group with multiplication defined by . Let A be a nonzero semiprime ring.

(i). The semigroup ring is a subdirect product of copies of A.

is semiprime.

(iii). If Q is a prime ideal of A(S), then is a prime ideal of A and

(iv). If I is a nonzero accessible subring of , then the cardinality |I| of I is at least κ.

Remark 3.1.[6] For every cardinal 1 \end{document} ]]> , the set of all finite words made from a (well-ordered) alphabet of cardinality lexicographically ordered, is a linearly ordered set satisfying the assumptions of Lemma 3.7.

Theorem 3.8.For every nonzero semiprime ring A and any cardinal | A | \end{document} ]]> , the radical is not a supernilpotent atom.

Proof. By Lemma 3.7, is a subdirect product of copies of A. This means that contains a family of ideals such that and the factor ring is isomorphic to A for every . If all then which is not the case. So there exists at least one such that Then which means that A is a nonzero homomorphic image of . This implies that since and radical classes are homomorphically closed. Consequently, . Moreover, because . So, if were a supernilpotent atom, then we would have which implies that . Then, it follows from Proposition 3.1 and Proposition 3.2 that contains a nonzero accessible subring , for some accessible subring S of A. But then which contradicts part (iv) of Lemma 3.7. □

Another example of a nonzero prime essential ring which does not generate a supernilpotent atom can be found in [8]. Namely,

Corollary 3.9.[8] Let Q be the field of all rational numbers. Then the set forms a semigroup with respect to the multiplication given by 1 . Let be the semigroup ring of the semigroup B over the two element . Then is not a supernilpotent atom. Consequently, is not an

In the next result, we refer to [20] and [21] to consider a concept from analysis, namely a Tychonof space, that is, a completely regular Hausdorf space. As a further example of a prime essential ring that does not generate a supernilpotent atom or a special atom, we obtain the following corollary.

Corollary 3.10.Let X be a Tychonof space that does not contain any isolated points. Define a set on is a continuous real-valued function defined on X} and let κ be any cardinal such that | C ( X ) | \end{document} ]]> . Then the radical is not a supernilpotent atom. Moreover, is not an N-atom.

Proof. It follows from Proposition 7 in [20] that C(X) is a prime essential ring. Thus C(X) is a semiprime ring. On the other hand, it follows from Theorem 3.8 that is not a supernilpotent atom and is not an N-atom. □

An example of Tychonof space can be seen in [20]. Furthermore, there exist prime rings R that generate supernilpotent atoms , as our next two examples show. Therefore, in general, is not a supernilpotent atom.

Example 3.11.Let F be a field and let be the ring of all matrices with entries from F. Then R is a nonzero simple prime ring and, it follows from that R generates a supernilpotent atom . Since every N-radical is supernilpotent, it therefore follows that . Moreover, is hereditary since every N-radical is left hereditary. But is not left hereditary since, for example, and the ring because its nonzero homomorphic image is not in Therefore,

Example 3.12.Let be the two-element field. As before, R generates the supernilpotent atom . Clearly, l is left and right hereditary since is a commutative ring and is left and right hereditary. However, since, otherwise, being an N-radical, would be left and right strong. Then, since and , the right strongness of implies that the ideal generated by is in Then, since , the left strongness of implies that the ideal generated by is in . But it is not so because, being a noncommutative prime ring, the ring is not isomorphic to . However, the radical , being an N-radical, is left and right strong.

In the proofs of the following two theorems (Theorem 3.13 and Theorem 3.14, we require Theorem 1 in [3], which states that if A is a nonzero ∗-ring, then is a supernilpotent atom. Furthermore, we also make use of Theorem 1 in [5], which asserts that if A is a nonzero ∗-ring, then is a special atom.

The next theorem shows the existence of a nonzero prime essential ring R such that the smallest N−radical containing R is an N−atom and the smallest special radical is a special atom.

Theorem 3.13.Let A be a nonzero semiprime ring and let , where is the set of all finite words made from a (well-ordered) alphabet of cardinality κ. If A is then the smallest N−radical containing R is an N−atom and the smallest special radical containing R is a special atom.

Proof. It follows from the proof of Theorem 3.8 that . Therefore, . It follows from Theorem 1 in [3] that is a supernilpotent atom. Hence, . In the other words, is a supernilpotent atom. Furthermore, it follows from Theorem 3.5 that is an

On the other hand, arguing as in the proof of Theorem 3.8, we get which implies that . Now, since A is a nonzero ∗−ring, it follows from Theorem 1 in [5] that is a special atom and where π is the class of all prime rings, is the essential closure of the hereditary closure of and is the upper radical determined by the class Thus, since part (iii) of Lemma 3.7 guarantees that every nonzero prime homomorphic image of is isomorphic to some nonzero prime homomorphic image of A and the only nonzero prime homomorphic image of A is A because A is a nonzero ∗−ring. Consequently, and so □

The existence of a nonzero prime essential ring that generates a special atom, a supernilpotent atom, and an shown in Theorem 3.13 shows that Question 1 posed in [8] has a positive answer. The following theorem describes a prime essential ring that does not generate a supernilpotent atom and a special atom.

Theorem 3.14. Let Z be the ring of integers. For every cardinal | \mathbb { Z } | \end{document} ]]> the special radical is not a special atom and the supernilpotent radical is not a supernilpotent atom.

Proof. Arguing as in the proof of Theorem 3.8, we get which implies that . Moreover, since for every prime number it follows that . But, since is a ∗−ring, it follows from Theorem 1 in [5] that . This implies that as otherwise, we would have and then for a prime number which is not the case as . Thus which clearly shows that is not a special atom. □

11

Our final result describes a case when the converse of Theorem 3.5 does hold. For that, we need the following two observations:

Lemma 3.15.Let A and T be nonzero semiprime commutative rings. Then, if , then

Proof. Clearly, for every commutative ring if and only if . Hence, if is a semiprime commutative ring such that , then for every nonzero semiprime homomorphic image T of T. Then, it follows from Theorem , that contains a left ideal such that , for some . But, since both rings T and A are commutative, this means that and . Moreover, since is semiprime. In other words, every nonzero semiprime homomorphic image of contains a nonzero ideal , for some accessible subring S of .

which means that because

Lemma 3.16.[4] Let L be a left ideal of a ring R and a and L is a commutative ring, then R is also commutative.

Let A be a nonzero commutative semiprime ring. As the final result of this paper, we give a necessary and suficient condition for the smallest N−radical containing A to be an N-atom in the following theorem.

Theorem 3.17.Let A be a nonzero commutative semiprime ring. Then is a supernilpotent atom if and only is an N-atom.

Proof. Assume that A is a nonzero commutative semiprime ring such that is an N-atom. Let be a nonzero prime ring. Then, T contains a nonzero ideal and since is prime. Moreover, I is commutative because every ring in HIR is commutative since R is commutative. it follows from Lemma 3.16 that T is a commutative ring. Moreover, since , it follows that . Then which gives , since is an N-atom. This shows that . But, as both rings and A are commutative, this, in view of Lemma 3.15, means that . Consequently, . Moreover, since Hence which, in view of Lemma 3.4 (v), means that is a supernilpotent atom.

The remaining part of the proof follows directly from Theorem 3.5. □

Based on the results that are explained in the previous section, we may infer that there exist prime essential rings which do not generate N−atoms. On the other hand, the existence of the ∗−ring which was introduced in [5] is important since we can construct a prime essential ring which generates an N−atom as shown in Theorem 3.13. Furthermore, let A be a nonzero semiprime ring. As the main result of this research, we infer that the smallest N−radical containing A is an N-atom if the smallest supernilpotent radical containing A is a supernilpotent atom.

Conversely, if the smallest containing A is an N-atom, we can conclude that the smallest supernilpotent radical containing A is a supernilpotent atom. Furthermore, the smallest special radical containing A is a special atom. This gives a positive answer for Question 1 posed in [8].