EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6011 ISSN 1307-5543 – ejpam.com Published by New York Business Global NJ-Abelian Rings: an Abelian-Like Approach Muhammad Saad1, Samia M. Abdelwahab2,3,∗ 1 Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, 21511 Alexandria, Egypt 2 Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, 11790 Helwan, Egypt 3Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, 51452 Buraidah, Saudi Arabia Abstract. This article extends the concept of NJ-semicommutative rings to introduce the broader class of NJ-abelian rings, which are defined by properties involving nilpotent elements and the Jacobson radical. We investigate the unique algebraic properties of NJ-abelian rings and analyze their relationships with various types of rings, including abelian, reduced, J-clean, local, and Dedekind-finite rings. In particular, we show that every NJ-semicommutative ring is NJ-abelian, much like every semicommutative ring is abelian. 2020 Mathematics Subject Classifications: 16U80, 16U99, 16U40 Key Words and Phrases: NJ-abelian, J-abelian, NJ-semicommutative, J-reduced 1. Introduction Though R is an associative ring with an identity, J(R) is the Jacobson radical of R, and N(R) is the set of nilpotent elements of R. A ring R is called semiprimative if J(R) = 0 and reduced if N(R) = 0. An idempotent e of a ring R is said to be left (resp. right) semicentral if (1−e)Re = 0 (resp. eR(1−e) = 0). If an idempotent e is both left and right semicentral, then it is central. A ring R is abelian if all of its idempotents are central. A ring R is called J-abelian if ae− ea ∈ J(R) for all e2 = e, a ∈ R. Abelian rings are easily shown to be J-abelian, but the converse is not true in general (see [1, 2]). Recall [3], a ring R is said to be semicommutative if ab = 0 implies aRb = 0 for any a, b ∈ R. The concept of semicommutative rings has been introduced in other terms in [4–6]. In [7], Subba and Subedi investigated a new class of rings called NJ-semicomutative. These rings generalize the notion of semicommutative rings by exploring the relation- ship between nilpotent elements and the Jacobson radical. A ring R is said to be NJ- semicommutative if aRb ⊆ J(R) wherever ab ∈ N(R) for all elements a, b ∈ R. This ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6011 Email addresses: m.saad@alexu.edu.eg (M. Saad), sam.mahmoud@qu.edu.sa (S. M. Abdelwahab) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 2 of 13 allows for a deeper understanding of the structure of these rings, particularly how the properties of nilpotent elements influence the behavior of the ring as a whole. In the same context, we define the NJ-abelian property. By introducing the concept of NJ-abelian rings, we aim to define a class of rings that incorporates both nilpotent elements and the Jacobson radical, potentially exploring the interactions between these two sets and the properties that result from them. Investigating the structure of NJ-abelian rings may reveal intriguing characteristics and properties that either align with or contrast with those found in NJ-semicommutative rings. You might explore questions such as • How do NJ-abelian rings relate to traditional abelian rings? • What conditions define a ring as NJ-abelian? • Are there any specific examples or counterexamples that illustrate the behavior of these rings? In this article, the following notations are used for a ring R: I(R) for the set of idempotents of R, N2(R) for the set of all square-zero elements of R (i.e., the nilpotent elements of index 2 or 1), U(R) for the set of units of R, and Mn(R) (resp. T n(R)) for the ring of all matrices (resp. upper triangular matrices) over R. 2. Basic Results We will talk about the NJ-abelian concept, come up with some basic results, and show how it is related to other concepts like the J-abelian, NJ-semicommutative, and J-reduced conditions. Definition 1. A ring R (not necessarily with identity) is called NJ-abelian if e2 = e, a ∈ R, and ae ∈ N(R), then aRe ⊆ J(R). The next proposition gives an equivalent condition for J-abelianity of rings, which will be used to show that every NJ-abelian ring is J-abelian. Proposition 1. A ring R is J-abelian if and only if ef ∈ N(R) for any e, f ∈ I(R) implies eRf ⊆ J(R). Proof. (⇐) Let e be an idempotent of R. Then e(1 − e) = 0 ∈ N(R), and hence eR(1 − e) ⊆ J(R). Similarly, eR(1 − e) ⊆ J(R). So, for every r ∈ R, we have er − re = er(1− e) + (1− e)(−r)e ∈ eR(1− e) + (1− e)Re ⊆ J(R) and R is J-abelian. (⇒) Let ef ∈ N(R) for some idempotents e and f of R of nilpotency index n. Define the idempotent g = 1−f+(1−f)erf for arbitrary r in R. So, (1−f)erf = fg−gf ∈ J(R) and erf−eferf = e(erf−ferf) = e(1−f)erf ∈ J(R). Applying the J-abelian condition, we get fer − fefer ∈ J(R). Beginning with the inclusion and sequentially multiplying the element by ef from left to right, we obtain( fer − (fe)2r ) + ( (fe)2r − (fe)3r ) + ( (fe)3r − (fe)4r ) + · · · ( (fe)n−1r − (fe)nr ) ∈ J(R) M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 3 of 13 and fer ∈ J(R) for every r ∈ R. From the J-abelianity of R, we get eRf ⊆ J(R). Corollary 1. Every NJ-abelian ring R is J-abelian. Here is a J-abelian ring that is not NJ-abelian. Example 1. Let R = Z4[x] and S = R/⟨x⟩. So, J(S) = ⟨2x⟩ and N(S) = ⟨2, x⟩. We have 1x ∈ N(S) while 1Sx ⊈ J(S). Thus, S is not NJ-abelian even though S is J-abelian. The next proposition shows that the definition of the NJ-abelian property is left-right symmetric. Proposition 2. Any ring R can satisfy the following equivalent conditions: (i) R is NJ-abelian; (ii) ae ∈ N(R) implies then eRa ⊆ J(R), where e2 = e, a ∈ R; (iii) ea ∈ N(R) implies then eRa ⊆ J(R), where e2 = e, a ∈ R; (iv) ea ∈ N(R) implies then aRe ⊆ J(R), where e2 = e, a ∈ R. Proof. (i)⇒(ii): If ae ∈ N(R), then aRe ⊆ J(R) from the NJ-abelianity of R. For every r ∈ R, we have era = e(ra) − (ra)e + rae ∈ J(R), since R is J-abelian from Proposition 1, and hence eRa ⊆ J(R). (ii)⇒(iii) is direct since ae is nilpotent if and only if ea is. (iii)⇒(iv): As in Proposition 1, one can prove that R is J-abelian. Now, are = (ar)e−e(ar)+ear ∈ J(R), for every e2 = e, r ∈ R by the hypotheses; that is, aRe ⊆ J(R). (iv)⇒(i) is direct again. Now, we aim to get a sufficient condition for the J-abelian ring to be NJ-abelian. According to [8], a ring R is called J-reduced if N(R) ⊆ J(R). In [9], a J-reduced ring is called an NJ ring. Obviously, every NJ-abelian ring R is J-reduced since all rings are with identity. The converse is not necessarily true, as the ring S in Example 1. Remind that every reduced ring is abelian. Hence, every reduced ring is NJ-abelian. The next proposition shows that J-reducedness is a sufficient and necessary condition for the J-abelian ring to become NJ-abelian. Proposition 3. A ring R is NJ-abelian if and only if it is J-reduced and J-abelian. Proof. The necessity is obvious. For sufficiency, assume that R is a J-reduced and J-abelian ring. Let ae ∈ N(R) for some e2 = e, a ∈ R. So, ea ∈ J(R) from the J- reducedness of R. For every r ∈ R, we have era = (e(ra)− (ra)e) + r(ae) ∈ J(R) since R is J-abelian. Thus eRa ⊆ J(R) and R is NJ-abelian. Nevertheless, it’s worth noting that NJ-abelian and abelian properties are distinct. The next examples show that the classes of NJ-abelian rings and abelian rings are independent. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 4 of 13 Example 2. Consider the ring R = {[ a b c d ] | a ≡ d mod 2, b ≡ c mod 2, a, b, c, d ∈ Z } . R has no nontrivial idempotents; consequently, R is abelian. The element α = [ 0 2 0 0 ] in R satisfies α1 ∈ N(R) while αR1 = [ 2Z 2Z 0 0 ] ⊈ J(R) = 0. Therefore, R is not NJ-abelian. Example 3. For any field F , the ring T 2(F ) is not abelian, where the set of nontrivial idempotents of R is {[ 1 a 0 0 ] , [ 0 a 0 1 ] | a ∈ F } is not central. For the idempotents of the form [ 0 a 0 1 ] , if αe ∈ N(R) for some α ∈ R, then α = [ b c 0 0 ] for arbitrary b, c ∈ F . So, αRe ∈ [ 0 F 0 0 ] = J(R). Therefore, R is NJ-abelian. The two examples above demonstrate that the abelian and NJ-abelian properties of rings are independent of each other. However, this does not imply that the abelian and NJ-abelian classes are completely disjoint. For instance, the ring Z is both abelian and NJ-abelian. Additionally, there exists a ring that is neither abelian nor NJ-abelian (see the next example). Example 4. Consider the ring R = F +F j, where F is a field, with 2−1 ∈ F , and j2 = 1. R is a commutative reduced ring, and its set of idempotents is {0, 1, 12(1 + j), 12(1 − j)}. Define the automorphism σ : R → R as σ(a + bj) = a − bj, for every a, b ∈ F . Let S = R[x;σ] be the skew polynomial ring with an indeterminate x over R. The element α = (1 + j)x in S satisfies α1 ∈ N(S) while αS1 ̸= 0. Therefore, S is not NJ-abelian since R is semiprimitive. Also, the nontrivial idempotents of S have the form e+(1± j)fx, where e2 = e, f ∈ R. So R is not abelian. The next proposition gives a sufficient condition to make an NJ-abelian ring abelian. Proposition 4. If R is a semiprimitive NJ-abelian ring, then R is abelian. Proof. For every idempotent e of a ring R, we have 0 = e(1 − e) ∈ N(R). So, eR(1− e) ⊆ J(R) = 0. Hence, e is right semicentral. Similarly, we demonstrate that e is also left semicentral. Therefore, e is central, and hence R is abelian. According to the definitions, every NJ-semicommutative ring is NJ-abelian. However, the converse does not necessarily hold, as illustrated by the following example. Example 5. Let K be a countable field. By [10, Lemma 3.7], there exists a nonzero nil algebra A over K such that A[x] has a zero upper nil radical. The ring R = (K + A)[x] is not NJ-semicommutative, as shown in [7, Example 4]. However, R is J-reduced, with N(R) = J(R) = A[x], and has only trivial idempotents. Therefore, R is NJ-abelian. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 5 of 13 Remind that a ring R is said to be Dedekind-finite if ab = 1 implies ba = 1 for every a, b ∈ R. The next proposition shows that every NJ-abelian ring is Dedekind-finite. Proposition 5. Every NJ-Abelian ring is Dedekind-finite. Proof. Let R be an NJ-Abelian ring with ab = 1 for some a, b ∈ R. So that (1− ba)2 = 1− 2ba+ baba = 1− ba and 1− ba is an idempotent of R. We have (1− ba)b = 0 ∈ N(R) and consequently 1− ba = (1− ba)ab ∈ (1− ba)Rb ⊆ J(R), from the NJ-Abelianity of R. Therefore, we have 1− ba = 0 and ba = 1, indicating that R is Dedekind-finite. In [11], an element a of a ring R is called left minimal if Ra is a minimal left ideal of R. Write MEl(R) to denote the set of all left minimal idempotents of R. A ring R is called left min-Abel if each left minimal idempotent is left semicentral. The next proposition states that a ring R is left min-Abel whenever it is NJ-Abelian. Proposition 6. Every NJ-abelian ring is left-min Abel. Proof. Assume e ∈ MEl(R) and r ∈ R. Define the element x = re − ere; hence, Rx ⊆ R. But Re is a minimal left ideal of R. So Rx = Re or x = 0. Indeed, xe ∈ N(R) and xRe ⊆ J(R), from the NJ-abelianity of R. If Rx = Re, then e = e2 ∈ RxRe ⊆ J(R) and e = 0; it is a contradiction. So, x = 0 and (1− e)Re = 0. Thus e is a left semicentral idempotent of R, and hence R left-min Abel. Remind that a ring R is called J-clean if for every a ∈ R, there exists an idempotent e ∈ R and b ∈ J(R) such that a = e+ b; that is, R = I(R) + J(R). We show that every J-clean ring is NJ-abelian in the following proposition. Proposition 7. Every J-clean ring R is NJ-abelian. Proof. Let R be a J-clean ring and a be a nilpotent element of R with index of nilpotency n. So, a = e+ b for some e ∈ I(R) and 0 = (e+ b)n ∈ e+ J(R) and e ∈ J(R). Therefore, e = 0 and a ∈ J(R); that R is J-reduced. Hence, R is NJ-abelian based on the results of [1, Lemma 2.4.] and Proposition 3. From the proposition above, we can get some corollaries that have been proved before in another work. Corollary 2 ([1], Lemma 2.4). Every J-clean ring is J-abelian. By Proposition 5, we also have the next corollary. Corollary 3 ([1], Theorem 2.10). Every J-clean ring is Dedekind-finite. Recall from [12] that the set of all elements of R that are nilpotent in R/J(R) is denoted by J#(R); that is, J#(R) = {a ∈ R | an ∈ J(R)}. It is obvious that both J(R) and N(R) are contained in J#(R). In [13], a ring R is called feckly reduced if R/J(R) is a reduced ring. The following proposition provides a more general result of [1, Proposition 2.6] under the same assumptions and shows that every feckly reduced ring is NJ-abelian. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 6 of 13 Proposition 8. If R is a feckly reduced ring, then R is NJ-abelian. Proof. From [13, Proposition 2.6], R satisfies J#(R). Now, let ea ∈ N(R) where e2 = e, a ∈ R. So, for every r ∈ R, we have (e − 1)are is nilpotent, and consequently ea, (e − 1)are ∈ J#(R) = J(R). So, are = (e − 1)are + eare ∈ J(R), for every r ∈ R, Thus, aRe ⊆ J(R), and hence R is NJ-abelian. Remind that a ring R is said to be local if it has only one maximal left (or right) ideal; equivalently, R/J(R) is a division ring. From Proposition 8 and [12, Lemma 1], we get directly the following result. Corollary 4. Every local ring is NJ-abelian. Here are interesting nontrivial implications in the class of rings with respect to the class of NJ-abelian rings. semicommutative reduced abelian J-clean NJ-semicommutative NJ-abelian J-abelian feckly reduced J-reduced Dedekind-finite 3. Extending of NJ-abelianity Note that the class of NJ-abelian rings is closed under direct products but not under closed subrings. Remember that the Nagata extension of a commutative ring R by an R-module M and an endomorphism σ of R is the ring of direct sum of the abelian groups R and M , with componentwise addition and multiplication defined as (r1,m1)(r2,m2) = (r1r2, σ(r1)m2 + m1r2) for all r1, r2 ∈ R and m1,m2 ∈ M . The next examples give an NJ-abelian subring of a ring that is not NJ . Example 6. Let R = Z⊕Z and σ be an automorphism on R defined as σ(a, b) = (b, a) for every (a, b) ∈ R. The Nagata extension of R by S and σ, denoted as S, is semiprimitive since R is reduced. Moreover, the idempotent ((1, 0), (0, 1)) and element ((0, 1), (0, 1)) of S satisfy ea ∈ N(S) while eSa = ((0, 0), (0,Z)) ⊈ J(S) = 0. Therefore, S is not NJ-abelian, while R is an NJ-abelian subring of S. Here is an NJ-abelian ring that has a non-NJ subring. Example 7. From [14, Example 4.8], let R = F ⟨x, y⟩ be a free algebra over a field F generated by the noncommuting indeterminates x and y. Consider the subring S = R/⟨y2⟩ of R. According to [9], S is not J-reduced, and consequently it is not NJ-reduced since S has identity. However, R is an NJ-abelian ring. Now, we give some results for subrings that are NJ-abelian due to the NJ-abelianity of their rings. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 7 of 13 Proposition 9. Let {Ri}i∈Λ be a class of rings for some index set Λ. Then ∏ i∈ΛRi is NJ-abelian if and only if Ri is NJ-abelian for every i ∈ Λ. Proof. The proof is routine. Corollary 5. Let R be a ring and e be a central idempotent of R. Then eR and (1− e)R are NJ-abelian if and only if R is NJ-abelian. Proposition 10. A ring R is NJ-abelian if and only if every corner of R is NJ-abelian. Proof. The sufficiency is trivial. For the necessity, let f2 = f, a ∈ eRe ∈ R, for some idempotent e of R such that af ∈ N(eRe) ⊆ N(R). So, a(eRe)f = aRf ∈ J(R) since R is NJ-abelian. But J(eRe) = eJ(R)e. So, a(eRe)f = eaRfe ⊆ eJ(R)e = J(eRe) and eRe is NJ-abelian. The next example shows that even if every corner eRe of a ring R is NJ-abelian for all nonidentity idempotents e, R is not necessarily NJ-abelian. Example 8. Let R = M2(Z) be the ring of all 2× 2 matrices over the ring of integers Z. Every nontrivial idempotent e of R satisfies eRe = Z, which is NJ-abelian, but R is not NJ-abelian since the elements α = [ 0 1 0 0 ] and ϵ2 = ϵ = [ 0 1 0 1 ] satisfy αϵ ∈ N(R) but αRϵ = [ 0 Z 0 0 ] . The following proposition shows that if a subring of an NJ-abelian ring is an ideal, then it is also NJ-abelian. Proposition 11. Every ideal of an NJ-abelian ring is NJ-abelian (as a ring without identity). Proof. Let R be an NJ-abelian ring and I be an ideal of R. Assume that ea ∈ N(I) where e2, a ∈ I. But N(I) ⊆ N(R) and R is NJ-abelian; hence, aRe ⊆ J(R). So, aIe ⊆ aRe ⊆ I ∩ J(R) = J(R) and I is NJ-abelian. Proposition 12. Let R be a ring such that R[x] is NJ-abelian. Then R is an NJ-abelian ring. Proof. Let e2 = e, a ∈ R such that ea ∈ N(R) ⊆ N(R[x]). From the NJ-abelianity of R[x], we have eRa ⊆ eR[x]a ∈ J(R[x]), and 1− eras is invertible in R[x] for all r, s ∈ R. But 1− eras ∈ R, and hence 1− eras ∈ U(R) for all r, s ∈ R. Thus R is NJ-abelian. It is natural to conjecture that R is an NJ-abelian ring if for any nonzero proper ideal I of R, R/I and I are both NJ-abelian rings, where I is considered a ring without identity. However, the following example provides a negative answer to this conjecture. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 8 of 13 Example 9. For a ring R, let S = M2(R), and I is the ideal generated by the commu- tators of S. Then S/I is commutative, and therefore it is NJ-abelian. The elements e = [ 1 1 0 0 ] and a = [ 1 1 −1 0 ] of S satisfy e2 = e and ea ∈ N(R). However, eSa = [ R R 0 0 ] ⊈ J(R). Thus, S is not NJ-abelian. However, if we take stronger independent conditions, such as “I is nil” and “I = J(R)”, then we may have an affirmative answer, as in the following. Proposition 13. For a ring R, if R/J(R) is NJ-abelian, then R is NJ-abelian. Proof. If e, f ∈ I(R) and ef ∈ N(R), then ēf̄ ∈ N(R/J(R)). But R/J(R) is NJ- abelian and ēr̄f̄ ∈ N(R/J(R)) = 0, for every r ∈ R. Thus, aRb ⊆ J(R), and R is NJ-abelian. Since every reduced ring is NJ-abelian, we have the following corollary. Corollary 6. If R/J(R) is reduced, then R is NJ-abelian. The converse of the previous proposition is not necessarily true, as shown in the next example. Example 10. Let S be the localization of Z at 3Z and R the set of quaternions over the ring S. According to [7], J(R) = 3R and R/J(R) = M2(Z3). Also, R is an NJ- semicommutative ring and consequently NJ-abelian. On the other hand, the idempotents e = [ 2 2 2 2 ] and f = [ 0 2 0 1 ] of R/J(R) satisfy ef ∈ N(R/J(R)) while e [ 1 1 2 1 ] f =[ 0 1 0 1 ] ̸∈ J(R/J(R)). Thus, R/J(R) is not NJ-abelian. Proposition 14. Let I be a nil ideal of R such that R/I is an NJ-abelian ring. Then R is NJ-abelian. Proof. Suppose that R/I is NJ-abelian and e = e2, a ∈ R such that ae ∈ N(R). Then ae ∈ N(R/I) and consequently a(R/I)e ⊆ J(R/I). Therefore, 1− ares ∈ U(R/I), for every r, s ∈ R. It means that 1 − (1 − ares)x ∈ I ⊆ N(R), for some x ∈ R. Thus (1 − ares)x is a unit in R, and hence 1 − ares has a right inverse for every r, s ∈ R. Therefore, aRe ⊆ J(R) and R are NJ-abelian. 4. Matrix extensions of NJ-abelian rings In this section, we study the NJ-abelian property for some ring extensions and their subrings. First, we show that the matrix ring Mn(R) over a ring R is not NJ-abelian for any ring R and n ≥ 2. Proposition 15. For any ring R and integer n ≥ 2, Mn(R) is not NJ-abelian. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 9 of 13 Proof. As shown in Example 9, M2(R) is not NJ-abelian for any ring R. From Proposition 10, every corner of an NJ-abelian ring is NJ-abelian. By induction, Mn(R) is not NJ-abelian for every n ≥ 2 since Mn(R) is a corner of Mn+1(R) for every n ≥ 1. While it is impossible to get a matrix ring satisfying the NJ-abelian condition, as shown in the example, we will explore the extent to which this property holds in some of its subrings or certain matrix contexts. For rings R and S, letM and N be (R,S)-bimodule and (S,R)-bimodule, respectively. The set of all matrices of the form [ r m n s ] , where r ∈ R, s ∈ S, m ∈ M , and n ∈ N . Using the standard matrix addition, we can identify this as an abelian group. To define a matrix multiplication for these elements, we need to define the products of mn and nm in R and S, respectively, for every m ∈ M and n ∈ N . We assume that there are two bimodule homomorphisms ϕ : (M,N) → R and ψ(N,M) → S. Simplify mn = ϕ(m,n) and nm = ψ(n,m) for all m ∈ M and n ∈ N . These maps satisfy the associativity conditions that are required to make the set with usual matrix addition and context multiplication an associative ring with identity, notated by [ R M N S ] . This ring is called the Morita context (R,M,N, S, ϕ, ψ), or a formal matrix ring (of order 2), or a ring of generalized matrices. The readers are referred to [15–19] as well as the references there for detailed information on the study in Morita contexts. Recall [20], a Morita context T = [ R M N S ] is called trivial if MN = 0 and NM = 0. The following lemma describes the idempotents, nilpotent elements, and Jacobson radicals in a trivial Morita context. Lemma 1. For two rings, R and S, let T = [ R M N S ] be a trivial Morita context. Then we have the following descriptions: (i) I(T ) ⊆ [ I(R) M N I(S) ] ; (ii) N(T ) = [ N(R) M N N(S) ] ; (iii) J(T ) = [ J(R) M N J(S) ] . Proof. The proofs of (i) and (ii) are straightforward, while (iii) is obtained directly from [21, Lemma 3.1]. Proposition 16. Suppose that T = [ R M N R ] is a trivial Morita context. Then R is NJ-abelian if and only if M and N are NJ-abelian. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 10 of 13 Proof. The sufficiency is straightforward from Proposition 10. For the necessity, assume that both M and N are NJ-abelian. Let ϵ = [ e m n f ] be an idempotent of T and α = [ a x y b ] be an arbitrary element of T . So, e ∈ I(R) and f ∈ I(S), from the previous lemma. If ϵα ∈ N(T ), then ea ∈ N(R) and fb ∈ N(S). So, eRa ⊆ J(R) and fRb ⊆ J(S) from the NJ-abelianity of R and S. Now, ϵTα = [ eRa eRx fNy fSb ] ⊆ [ J(R) M N J(S) ] = J(T ) and T is an NJ-abelian ring. Notice that formal triangular matrix rings are obvious examples of trivial Morita con- texts. So, we have the following corollary. Corollary 7. Let M represent an (R,S)-bimodule for the rings R and S. Then T =[ R M 0 S ] is NJ-abelian if and only if both R and S are NJ-abelian. In [22], if R is a ring and M is an (R,R)-bimodule, then the direct sum of abelian groups R and M with standard addition and multiplication defined as (r1,m1)(r2,m2) = (r1r2, r1m2 + m1r2), for all r1, r2 ∈ R, m1,m2 ∈ M , is a ring with the identity (1, 0). This ring is the trivial extension of R by M , denoted by T (R,M). Notice that the trivial extension of R by M is that the formal triangular matrix ring [ R M 0 R ] . So, we have the following corollary. Corollary 8. LetM represent an (R,R)-bimodule associated with a ring R. Then T (R,M) is NJ-abelian if and only if R is NJ-abelian. Proposition 17. For any ring R, the following conditions are equivalent. (i) R is NJ-abelian; (ii) T n(R) is NJ-abelian for some n > 1; (iii) T n(R) is NJ-abelian for every n > 1. Proof. (i)⇒(ii): If R is an NJ-abelian, then T 2(R) is a formal triangular matrix, and consequently it is NJ-abelian from Corollary 7. (ii)⇒(iii): Applying the mathematical induction on n, assume that T n(R) is NJ-abelian for some n > 1; then R is NJ-abelian by Proposition 10. Notice that T n+1(R) is the formal triangular matrix [ T n(R) Rn 0 R ] , where Rn is the (T n(R), R)-bimodule of n-by-1 matrices over R. Therefore T n+1(R) is NJ-abelian by Corollary 7. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 11 of 13 For a ring R, define a subring Sn(R) of T n(R) as Sn(R) =   a a12 a13 a14 · · · a1n 0 a a23 a24 · · · a2n 0 0 a a34 · · · a3n 0 0 0 a · · · a4n ... ... ... ... . . . ... 0 0 0 0 · · · a  |a, aij ∈ R  , where n ≥ 2 is a positive integer. Also, we have a subring of Sn(R) defined below: V n(R) =   a1 a2 a3 a4 · · · an 0 a1 a2 a3 · · · an−1 0 0 a1 a2 · · · an−2 0 0 0 a1 · · · an−3 ... ... ... ... . . . ... 0 0 0 0 · · · a1  |ai ∈ R  , where n ≥ 2 is a positive integer. The next proposition shows that the NJ-abelianity of a ring R, Sn(R), and Vn(R) are equivalent for every n. Proposition 18. For any ring R, the following conditions are equivalent. (i) R is NJ-abelian; (ii) Sn(R) is NJ-abelian for some n > 1; (iii) Sn(R) is NJ-abelian for every n > 1; (iv) V n(R) is NJ-abelian for some n > 1; (v) V n(R) is NJ-abelian for every n > 1. Proof. (i)⇒(iii): For every n ≥ 2, consider the ideal In(R) of Sn(R) consisting of all elements of Sn(R) with zero diagonal entries. Notice that I is a nil ideal and R ∼= Sn(R)/In(R). Thus, Sn(R) is NJ-abelian from Proposition 14. (iii)⇒(i) is direct from Proposition 10. (i)⇔(v) may be proved by the same technique of proving (i)⇔(iii). (i)⇔(ii)⇔(iv) is obtained directly from Corollary 8 since T (R,R) = S2(R) = V 2(R). Corollary 9. A ring R is NJ-abelian if and only if R[x]/⟨xn⟩ is NJ-abelian for any positive integer n. M. Saad, S. M. Abdelwahab / Eur. J. Pure Appl. Math, 18 (2) (2025), 6011 12 of 13 Conclusion In this article, we introduced and studied a new class of rings, called NJ-abelian rings, defined by a condition that connects idempotents and nilpotent elements through the Jacobson radical. We showed that this class properly extends NJ-semicommutative rings, analogous to how abelian rings extend semicommutative rings. We proved that every NJ-abelian ring is J-abelian and that the NJ-abelian condition is symmetric with respect to left and right multiplication. Several equivalent characteriza- tions were established, notably that a ring is NJ-abelian if and only if it is both J-reduced and J-abelian. Moreover, we demonstrated that many classical ring classes, such as J- clean rings, local rings, and feckly reduced rings, are NJ-abelian, and we clarified their implications with suitable counterexamples. In addition, we examined the behavior of the NJ-abelian property under various ring constructions and extensions. We showed that NJ-abelianity is not generally preserved in full matrix rings or certain direct sums and ideals, whereas it can hold in specific subrings like upper triangular matrix rings and some of their substructures. The results presented here contribute to a deeper understanding of the interaction between nilpotent elements, idempotents, and the Jacobson radical. 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