Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 48, No. 1 (2026) pp. 60–64.  
ON WEAKLY NIL CLEAN AND WEAKLY NIL  
NEAT RINGS  
Santosh Kumar Pandey  
Abstract. In this paper we obtain some significant results on weakly nil clean and  
weakly nil neat rings. In addition this work extends some results on commutative weakly  
nil clean and weakly nil neat rings appeared in [4] to noncommutative weakly nil clean rings  
and weakly nil neat rings. More interestingly we provide certain results on noncommutative  
weakly nil neat rings having more than two idempotents.  
Keywords: Nil clean ring, weakly nil-clean ring, weakly nil neat ring, weakly UU ring.  
2010 AMS Subject Classification: 13B99, 16E50,16U99.  
1. Introduction  
Throughout this paper, R is an associative ring with identity element. We denote the  
set of all units, set of all idempotents and the set of all nilpotents in R by U(R), Id(R)  
and N(R) respectively. The Jacobson radical of R is denoted by J(R) and the set of all  
unipotent elements in R is denoted by Unip(R).  
McGovern [1] has introduced the notion of neat rings which is defined as a ring whose  
every proper homomorphic images is clean. M. Samiei has studied nil neat rings in [2] and  
these rings are defined as rings whose every proper homomorphic images are nil clean. We  
have studied and extended some results presented in [2] on nil neat rings in [3]. Danchev  
and Samiei have studied the notion of weakly nil neat rings in [4]. A ring R is called a  
weakly nil neat ring if every proper homomorphic image of R is weakly nil clean [4].  
In [4] commutative weakly nil clean rings and weakly nil neat rings have been studied.  
Following [4] here we obtain some significant results on weakly nil clean and weakly nil  
neat rings and we extend some results of [4]. This should be emphasized that in [4] each  
ring is a commutative ring however in the present paper a ring R is not necessarily a  
commutative ring.  
For the sake of convenience we shall provide some useful definitions needed to under-  
stand this paper as follows. A ring R is called a clean ring if every element in R is the sum  
of a unit and an idempotent [10-11]. A ring R is called a nil clean ring if every element  
of R is the sum of a nilpotent and an idempotent [6]. A ring R is called a weakly nil  
clean ring if for each a ∈ R we have a = u + v or a = u − v for some u ∈ N(R) and  
v ∈ Id(R) [9]. A ring R is called weakly UU ring if for each a ∈ U(R) we have a = n + 1  
or a = n − 1 [4]. Here n ∈ N(R). In other words a ring R is called weakly UU ring if for  
each a ∈ U(R) we have either a ∈ Unip(R) or a ∈ −Unip(R).  
In the next section we provide main results of this paper.  
60  
ON WEAKLY NIL CLEAN AND WEAKLY NIL NEAT RINGS  
61  
2. Main Results  
We obtain the following results on weakly nil clean and weakly nil neat rings. It may  
be emphasized that these results hold for commutative as well as noncommutative rings  
under the given condition however in [4], all results have been obtained for commutative  
rings only.  
Proposition 2.1. Let R is a ring without non-trivial idempotents. Then R is a weakly  
nil-clean ring iff for each a ∈ R, either a ∈ N(R), or a ∈ Unip(R) or a ∈ −Unip(R).  
Proof. Let R is a ring without non-trivial idempotents. Let R is a weakly nil-clean ring  
and a is any element of R. Clearly a = b+c or a = b−c such that b ∈ Id(R) and c ∈ N(R).  
Clearly if b = 0, then a ∈ N(R) and if b = 1, then a ∈ Unip(R) or a ∈ −Unip(R). Hence  
for each a ∈ R, either a ∈ N(R), or a ∈ Unip(R) or a ∈ −Unip(R). Conversely let for  
each a ∈ R, either a ∈ N(R), or a ∈ Unip(R) or a ∈ −Unip(R). Let a ∈ N(R), then  
a = 0 + a. If a ∈ Unip(R), then a = 1 + c for some c ∈ N(R). If a ∈ −Unip(R), then  
a = c − 1 for some c ∈ N(R). Hence R is a weakly nil clean ring.  
Proposition 2.2. Let R is a ring without non-trivial idempotents. Then the following  
results are equivalent.  
(i) R is a weakly nil clean ring such that Unip(R) = −Unip(R).  
J(R)  
R
(ii) J(R) is nil and  
is the field of order three.  
Proof. (i) ⇔ (ii). Let R is a weakly nil clean ring such that Unip(R) = −Unip(R).  
Clearly for each a ∈ R, either a ∈ N(R), or a ∈ Unip(R) or a ∈ −Unip(R). This implies  
that if a ∈ R, then either a is nilpotent or a is a unit. Thus all non-unit is a nilpotent  
element. This leads that u + v ∈ N(R) and uv ∈ N(R) for each u, v ∈ N(R). Also  
au ∈ N(R), ua ∈ N(R) for each a ∈ R and u ∈ N(R). Thus N(R) a nil ideal of R and  
so J(R) = N(R) as J(R) cannot contain any element a ∈ R such that a ∈ Unip(R) or  
a ∈ −Unip(R).  
Further let a ∈ R is any element of R. If a ∈ J(R), then a + J(R) = J(R). Similarly  
if a ∈ Unip(R), then we have a + J(R) = 1 + J(R) since 1 + J(R) = Unip(R). Next  
if a ∈ −Unip(R), then we have a + J(R) = −1 + J(R) since −1 + J(R) = −Unip(R).  
J(R)  
R
Hence  
= {J(R), 1 + J(R), −1 + J(R)} is the field of order three. The converse easily  
follows.  
Corollary 2.3. Let R is a ring without non-trivial idempotents. Then the following results  
are equivalent.  
(i) R is a weakly nil clean ring such that Unip(R) = −Unip(R).  
J(R)  
R
(ii) J(R) is nil and  
is the field of order two.  
Remark 2.4. Let R is a weakly nil clean ring such that Unip(R) = −Unip(R). Then  
for each a ∈ N(R), we have some b ∈ N(R) such that 1 + a = b − 1. This implies that  
2 = b − a. As already noted N(R) is an ideal, so 2 = b − a ∈ N(R). Thus 2 is nilpotent.  
62  
Santosh Kumar Pandey  
Proposition 2.5. Let R is a ring without non-trivial idempotents. If R is a weakly nil  
clean ring, then R has a unique maximal ideal.  
Proof. Let R is a ring without non-trivial idempotents. Let R is a weakly nil clean ring.  
We have noted above (we refer the proof of the Proposition 2.4) that J(R) = N(R) is  
a nil ideal of R. Let I is any ideal of R. Since for each a ∈ R, either a ∈ N(R), or  
a ∈ Unip(R) or a ∈ −Unip(R) and so each element of I must be nilpotent. Hence I must  
be contained in J(R). Hence J(R) is the unique maximal ideal of R.  
Proposition 2.6. Let R is a ring without non-trivial idempotents. Then the following  
results are equivalent.  
(i) R is a local weakly nil clean ring.  
(ii) R is an indecomposable weakly nil clean ring.  
(i) ⇔ (ii). Let R is a ring without non-trivial idempotents. Let R is a local weakly nil  
clean ring. This immediately follows that R is an indecomposable weakly nil clean ring.  
Conversely let R is an indecomposable weakly nil clean ring. It suffices to prove that R  
is a local ring. It is known that a weakly nil clean ring is a clean ring ([9, Corollary 8]).  
Therefore R is a clean ring. Further a clean ring R without non-trivial idempotents is a  
local ring [10]. Hence R is a local weakly nil clean ring as desired.  
Proposition 2.7. Let R is a ring without non-trivial idempotents. Then the following  
results are equivalent.  
(i) R is a weakly nil clean ring.  
(ii) R is a Weakly UU ring and R has exactly one prime ideal.  
Proof. (i) ⇒ (ii) Let R is a weakly nil clean ring without non-trivial idempotents.  
Therefore for each a ∈ R, either a ∈ N(R), or a ∈ Unip(R) or a ∈ −Unip(R) (using  
Proposition 2.1). Thus R is a weakly UU ring by the definition of UU ring. Now we shall  
prove that R has exactly one prime ideal. Let I is a prime ideal of R. It is easy to note  
that for each a ∈ R there exists a positive integer n such that anR is a two sided ideal  
(since an = 1 or an = 0 for some positive integer n ). This implies that R is weakly  
right duo ring (we refer [5]). Since R is without nontrivial idempotents, therefore R is an  
abelian weakly nil clean ring and every abelian weakly nil clean ring is strongly π-regular  
(Proposition 15, [9]). Now using Theorem 3.10 [5] we obtain that I is a maximal ideal.  
But we have already proved that R has a unique maximal ideal (see Proposition 2.5).  
Hence I = N(R) is the only prime ideal of R. (ii) ⇒ (i) easily follows as if R is a UU ring  
then for each non-nilpotent (unit) element a ∈ R we have a ∈ Unip(R) or a ∈ −Unip(R)  
and if a ∈ R is non-unit then a ∈ N(R). Hence for each a ∈ R, either a ∈ N(R) or,  
a ∈ Unip(R) or a ∈ −Unip(R). Therefore by Proposition 2.1R is a weakly nil clean ring.  
Corollary 2.8. Let R is a ring without non-trivial idempotents. Then the following  
statements are equivalent.  
(i) For each a ∈ R, either a ∈ N(R) or, a ∈ Unip(R) or a ∈ −Unip(R).  
ON WEAKLY NIL CLEAN AND WEAKLY NIL NEAT RINGS  
63  
(ii) R is a weakly UU ring and R has exactly one prime ideal.  
Proposition 2.9. Let R is a non-reduced ring such that K(R) = N(R). Here K(R) is  
the set of all elements a ∈ R satisfying a = kb for some fixed positive integer k and some  
b ∈ R. Then R is weakly nil neat iff R is weakly nil clean.  
Proof. Let R is a non-reduced ring such that K(R) = N(R). Here K(R) is the set of  
all elements a ∈ R satisfying a = kb for some fixed positive integer k and some b ∈ R.  
Further let R is a weakly nil neat ring. Since K(R) = N(R) and therefore it follows from  
R
N(R)  
Proposition 2.7[7] that N(R) is an ideal of R. Clearly  
is a weakly nil clean ring  
because every proper homomorphic image of R is a weakly nil clean ring. Hence R is  
a weakly nil clean ring (we refer Lemma 1, [9]). Conversely let R is a weakly nil clean  
ring. Then each homomorphic image of R is weakly nil clean (we refer Lemma 1, [9]).  
Therefore it follows that R is weakly nil neat ring.  
Proposition 2.10. Let R is a non-reduced ring such that K(R) = I. Here K(R) is the  
set of all elements a ∈ R satisfying a = kb for some fixed positive integer k and some  
b ∈ R and I is a non-zero subset of N(R). Then R is weakly nil neat iff R is weakly  
nil clean.  
Proof. The proof of this Proposition is similar to the proof of above Proposition 2.9.  
R
I
More explicitly like above in this case  
is a weakly nil clean ring and therefore it follows  
that R is a weakly nil clean ring. Conversely if R is a weakly nil clean ring, then every  
homomorphic image of R is a weakly nil clean ring and hence R is a weakly nil neat ring.  
Proposition 2.11. Let R is a non-reduced ring such that E(R) = N(R), where E(R) is  
the set of all even elements of R. Then R is weakly nil neat iff R is weakly nil clean.  
Proof. The proof can be produced using arguments given in the proof of Proposition 2.9  
and noting that N(R) is an ideal (we refer Proposition 2.6, [7]). More explicitly like above  
R
N(R)  
is a weakly nil clean ring and therefore it follows that R is a weakly nil clean ring.  
Conversely if R is a weakly nil clean ring, then every homomorphic image of R is a weakly  
nil clean ring and hence R is a weakly nil neat ring.  
Proposition 2.12. Let R is a non-reduced ring such that E(R) = I, where E(R) is the  
set of all even elements of R and I is a non-zero subset of N(R). Then R is weakly nil  
neat iff R is weakly nil clean.  
Proof. The proof is similar to the Proof of proposition 2.11. For even elements one may  
refer [8].  
Remark 2.13. A reduced weakly nil clean ring is always commutative (Corollary 14, [9]).  
Conclusion  
Unlike [4] our results presented here hold for commutative as well as noncommutative  
weakly nil clean rings and weakly nil neat rings. Thus we have improved some results  
64  
Santosh Kumar Pandey  
on weakly nil clean rings and weakly nil neat rings presented in [4]. In addition we have  
given some interesting results on noncommutative weakly nil neat rings having nontrivial  
idempotents.  
Though we have given certain results on noncommutative weakly nil neat rings having  
non-trivial idempotents however in future a general theory of noncommutative weakly nil  
neat rings may be developed and such a theory appears to be highly desirous and the  
results presented here may motivate for the same.  
3. Statements and Declaration  
The author declares that there is no competing interest and there was no funding  
support available for this research.  
4. Acknowledgements  
The author is grateful to A. Pandit for his help.  
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50 (1)(2020), 37-44.  
[3] Pandey, S. K. On rings whose proper homomorphic images are nil clean in P. Saikia, D. J. Mahanta,  
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[4] Danchev, P. Samiei, M. Commutative weakly nil neat rings, Novi Sad J. Math., 50 (2) (2020), 51-59.  
[5] Gorman, A. B. Diesl, A. Ideally nil clean rings, Commun Algebra, 49 (11) (2021), 4788-4799.  
[6] Diesl, A. J. Nil clean rings, J. Algebra, 383 (2013), 197-211.  
[7] Pandey, S. K. A note on the Koethe Conjecture, Aligarh Bull. Math., 41 (2) (2022), 29-33.  
[8] Pandey, S. K. Nil elements and even square rings, International Journal of Algebra, 11(1) (2017),  
1-7.  
[9] Breaz, S., Danchev, P., Zhou, Y. Rings in which every element is either a sum or a difference of a  
nilpotent and an idempotent, J. Algebra Appl. 15, 8 (2016), 1650148.  
[10] Nicholson, W. K., Zhou, Y. Rings in which elements are uniquely the sum of an idempotent and a  
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[11] Nicholson, W. K., Zhou, Y. Clean rings: a survey, Advances in Ring Theory, World Sci. Publ.,  
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(Received, July 31, 2025)  
(Revised, October 3, 2025)  
Faculty of Science, Technology and Forensic,  
Sardar Patel University of Police, Security and Criminal Justice,  
Lordi Pandit Ji-342037, Jodhpur, India.  
Email: skpandey12@gmail.com