Nil-quasi-clean companion matrices
Let \(R\) be a ring with identity. An element \(e\) in \(R\) is called a quasi-idempotent element if \(e^2=ke\) for some central unit \(k\) in \(R\). For an element \(b\) in \(R\), if there is a positive integer \(m\) such that \(b^m=0\), then \(b\) is called a nilpotent element of \(R\). An elemen...
Збережено в:
| Дата: | 2026 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2026
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2466 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
| Завантажити файл: |  |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1870196970128146432 |
|---|---|
| author | Su, Huadong Liu, Shifeng |
| author_facet | Su, Huadong Liu, Shifeng |
| author_institution_txt_mv | [
{
"author": "Huadong Su",
"institution": "Beibu Gulf University"
},
{
"author": "Shifeng Liu",
"institution": "Guangxi University"
}
] |
| author_sort | Su, Huadong |
| baseUrl_str | https://admjournal.luguniv.edu.ua/index.php/adm/oai |
| collection | OJS |
| datestamp_date | 2026-07-08T07:55:33Z |
| description | Let \(R\) be a ring with identity. An element \(e\) in \(R\) is called a quasi-idempotent element if \(e^2=ke\) for some central unit \(k\) in \(R\). For an element \(b\) in \(R\), if there is a positive integer \(m\) such that \(b^m=0\), then \(b\) is called a nilpotent element of \(R\). An element \(r\) in \(R\) is called a nil-quasi-clean element if \(r\) is a sum of a quasi-idempotent and a nilpotent. If every element of \(R\) is nil-quasi-clean, then \(R\) is called a nil-quasi-clean ring. This paper completely determines nil-quasi-clean companion matrices over a field. |
| doi_str_mv | 10.12958/adm2466 |
| first_indexed | 2026-07-09T01:00:17Z |
| format | Article |
| fulltext |
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 41 (2026). Number 2, pp. 270–280
DOI:10.12958/adm2466
Nil-quasi-clean companion matrices
Huadong Su and Shifeng Liu
Communicated by A. Petravchuk
Abstract. Let R be a ring with identity. An element e in
R is called a quasi-idempotent element if e2 = ke for some central
unit k in R. For an element b in R, if there is a positive integer
m such that bm = 0, then b is called a nilpotent element of R.
An element r in R is called a nil-quasi-clean element if r is a sum
of a quasi-idempotent and a nilpotent. If every element of R is
nil-quasi-clean, then R is called a nil-quasi-clean ring. This paper
completely determines nil-quasi-clean companion matrices over a
field.
Introduction
Matrix ring is an important class of rings and it has many applications
in operation theory and others. This paper concerns the square matrices
over a field. Let us recall some definitions and notations in ring theory.
All rings we consider in this paper are associative with identity. Let
R be a ring. An element e in R is called an idempotent element if
e2 = e. For an element b in R, if there is a positive integer m such that
bm = 0, then b is called a nilpotent element of R. An element r in R is
The authors thank the reviewers for their valuable suggestions. This research was
supported by the National Natural Science Foundation of China (Grant No. 12261001,
12461001) and Research Foundation Ability Enhancement Project for Young and Mid-
dle aged Teachers in Guangxi Universities (2025KY0477).
2020 Mathematics Subject Classification: 16U40, 15B33, 16S50.
Key words and phrases: quasi-idempotent matrix, nil-quasi-clean, nilpotent
matrix, finite field.
https://doi.org/10.12958/adm2466
H. Su, S. Liu 271
called a nil-clean element if r is a sum of an idempotent and a nilpotent.
We use Id(R), Nil(R), U(R), NC(R) to denote the set of idempotents,
the set of nilpotent elements, the set of units, and the set of nil-clean
elements of R, respectively. If every element of R is nil-clean, that is,
NC(R) = R, then R is called a nil-clean ring. As usual, J(R), C(R)
denote the Jacobson radical and the center of a ring R, respectively. The
set of central units of R is Uc(R) = U(R) ∩ C(R). We also use char(R)
to denote the characteristic of R and R∗ = R\{0}. Recently, Tang et al.
introduced the definition of a quasi idempotent element in [10], which is
a generalization of idempotent element. Recall that an element a of R
is a quasi-idempotent if a2 = ka for some k ∈ Uc(R). We use QId(R) to
denote the set of all quasi-idempotents in R. An element r in R is called
a nil-quasi-clean element if r = a + b with a ∈ QId(R) and b ∈ Nil(R).
If every element of R is nil-quasi-clean, then R is called a nil-quasi-clean
ring.
The concept of nil-clean ring first appeared in [7], Diesl showed that
every nil-clean ring is clean. Many papers are devoted to the study
of clean rings and nil-clean rings, especially for the cleanness and nil-
cleanness of matrix ring, for examples, [1, 2, 6, 8]. Diesl in [7] posed an
open question that whether the matrix ring over a nil-clean is nil-clean.
Breaz et al. in [4] showed that the n × n matrix ring over a field F is
nil-clean ring if and only if F ∼= F2. For a division ring D, Kosan et
al. proved in [9] that Mn(D) is nil-clean if and only if D ∼= F2. In [5],
Cǎlugǎreanu proved that an invertible matrix inM2(Z) with trace 1 is nil-
clean. Concerning the nil-quasi-cleanness, Tang and Zhou in [11] showed
that the 2× 2 matrix ring over a division ring D is quasi-nil-clean if and
only if D is a perfect field with characteristic 2. They asked a question
that when the matrix ring Mn(R) over a division ring R is nil-quasi-
clean. Motivated by [3], in which the authors studied the nil-cleanness of
companion matrices, we study nil-quasi-cleanness of companion matrices
in this paper. Hoping it can be helpful and inspiring for characterizing
the nil-quasi-cleanness of general matrices.
1. Nil-quasi-clean companion matrices
In this section, we characterize the nil-quasi-cleanness of companion mat-
rices over a field. Most results are the generalization of those in [3]. Let
F be a field and n a positive integer. We denote
272 Nil-quasi-clean companion matrices
Cc0,c1,...,cn−1 =
0 0 · · · 0 −c0
1 0 · · · 0 −c1
0 1 · · · 0 −c2
· · · · · · · · · · · · · · ·
0 0 · · · 0 −cn−2
0 0 · · · 1 −cn−1
the n × n companion matrix, and the characteristic polynomial of the
companion matrix Cc0,c1,...,cn−1 is represented by
p(x) = Xn + cn−1X
n−1 + · · ·+ c1X + c0.
As we all known, any matrix can be put into Frobenius normal form, that
is, every square matrix A over a commutative ring is similar to a rational
canonical matrix. Since a matrix similar to a nilpotent(or idempotent)
matrix is still nilpotent(or idempotent), when we consider the nilpotency
or idempotency, it just consider the above companion matrix. Note that
a matrix similar to a quasi-idempotent matrix is also a quasi-idempotent
matrix.
For easy understanding, we first consider the low order matrices.
Firstly, consider a companion matrix of order 2.
Proposition 1. A companion matrix C =
(
0 −c0
1 −c1
)
over a finite field
F with char(F ) > 2 is nil-quasi-clean if and only if one of following
holds:
(1) c0 = c1 = 0;
(2) c0 = u2, c1 = −2u for some u ∈ F ∗;
(3) c1 = −u for some u ∈ F ∗.
Proof. ⇐ If c0 = c1 = 0, then C is clearly nilpotent and thus nil-quasi-
clean. If c0 = u2, c1 = −2u for some u ∈ F ∗, then we set C =
(
u 0
0 u
)
+(
−u −u2
1 u
)
= E+B, which is a nil-quasi-clean decomposition, as E2 =
uE ∈ QId(M2(F )) and B ∈ Nil(M2(F )). If c1 = −u for some u ∈ F ∗,
then we set C =
(
0 −c0
1 u
)
=
(
0 0
1 0
)
+
(
0 −c0
0 u
)
= B + E, which is
a nil-quasi-clean decomposition, as E2 = uE ∈ QId(M2(F )) and B ∈
Nil(M2(F )).
H. Su, S. Liu 273
⇒ We may assume that C = E+B with E2 = uE and B is nilpotent.
Then we have −c1 = trace(C) = trace(E) + trace(B) = trace(E) =
u · rank(E). If rank(E) = 0, then E = 0 and −c1 = 0. In this case, C is
nilpotent. So, we get c2 = 0 and hence (1) holds. If rank(E) = 1, then
we have c1 = −u and (3) happens. If rank(E) = 2, then E = uI2. As
C is nil-quasi-clean, then c1 = −2u and c0 = u2, which shows that (2)
happens.
Next, we consider the companion matrix of order 3.
Proposition 2. Let Cc0,c1,c2 be a companion matrix over a field F and
char(F ) = p. Then C is nil-quasi-clean if and only if one of following
holds:
(1) c2 ̸= 0;
(2) c2 = 0 and p = 2;
(3) c2 = 0, p = 3 and C = aI3 +B with B nilpotent;
(4) c2 = 0, p ≥ 5 and C is nilpotent.
Proof. ⇐ For the case (1), we let c2 = −u. Then we may decompose C
into
C =
−u −u2 + c1 −u3 + 2c1u
1 0 −c1
0 1 u
+
u u2 − c1 −c0 + u3 − 2c1u
0 0 0
0 0 0
= B + E. Note that B3 = 0 and E2 = uE. Thus C is nil-quasi-clean.
For the case (2), we have
C =
0 0 −c0
1 0 −c1
0 1 0
=
u 0 u3
1 u u2
0 1 0
+
u 0 −c0 − u3
0 u −c1 − u2
0 0 0
= B + E.
As B ∈ Nil(M3(F )) and E2 = uE ∈ QId(M3(F )), we know that C is
nil-quasi-clean. The cases (3) and (4) are clearly nil-quasi-clean.
⇒ Suppose that C = E + B, where E2 = uE and B is nilpotent.
Then we have−c2 = trace(C) = trace(E) = u·rank(E). If rank(E) ̸= 0,
then c2 = −u · rank(E), so (1) is correct. If rank(E) = 0, we proceed
with the characteristic of the field. When p = 2, C with nil-quasi-clean
decomposition, (2) is true. When p = 3, and C is nil-quasi-clean, we
274 Nil-quasi-clean companion matrices
have rank(E) = 0 or rank(E) = 3. Hence E = 0 or E = uI3 i.e.,
C = aI3 +B, a ∈ F , so (3) is true. When p ⩾ 5 and C is nil-quasi-clean,
we have E = 0. Hence C ∈ Nil(M3(F )), so (4) is also true.
Lemma 1 ([3, Lemma 1]). Let f = Xn + fn−1X
n−1 + · · ·+ f1X + f0 ∈
F [X] be a monic polynomial. For every (c1, · · · , cn−1) ∈ Fn−1 there
exists a unique tuple (α0, · · · , αn−1) ∈ Fn such that the matrix
M =
−αn−1 −αn−2 · · · −α1 −α0
1 0 · · · 0 −c1
0 1 · · · 0 −c2
...
...
. . .
...
...
0 0 · · · 0 −cn−2
0 0 · · · 1 −cn−1
∈ Mn(F )
is similar to the companion matrix Cf0,f1,...,fn−1 of f .
Next result is a generalization of [3, Proposition 3].
Proposition 3. Let n,m, k be three positive integers and n = m + k.
Fix c0, c1, . . . , cn−1 ∈ F and companion matrix Cc0,c1,...,cn−1. For each
polynomial g ∈ F [X] with deg(g) ⩽ n−2, there are two matrixes E,M ∈
Mn(F ) such that Cc0,c1,...,cn−1 = E+M , where E2 = uE for some u ∈ F ∗
and the characteristic polynomial of M is Xn + (k·u+ cn−1)X
n−1 + g.
Proof. Let g = fn−2X
n−2 + · · ·+ f1X + f0 ∈ F [X]. Consider the block
matrix
E =
(
uIk E12
0 0
)
∈ Mn(F ),
where E12 =
0 0 · · · 0 α0 − c0
0 0 · · · 0 α1 − c1
...
...
. . .
...
...
0 0 · · · 0 αk−2 − ck−2
αn−2 αn−3 · · · αk αk−1 − ck−1
∈ Mk×m(F ), and
α0, α1, · · · , αn−2 ∈ F .
By direct computation, we know that E2 = uE ∈ QId(Mn(F )). So
E is a quasi-idempotent matrix. To complete the proof, we show by
induction on k ⩾ 1 that there are uniquely determined α0, α1, · · · , αn−2
such that M = C−E has the characteristic polynomial f = Xn+(k·u+
cn−1)X
n−1 + g.
H. Su, S. Liu 275
The step k = 1 is Lemma 1 (note that fn−1 = αn−1 + cn−1), let
αn−1 = u and fn−1 = u+ cn−1. Suppose the claim is true for k ⩾ 2, and
let M = C − E. Expanding by the first row we get
|XIn −M | =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
X + u 0 · · · 0 0 · · · 0 α0
−1 X + u · · · 0 0 · · · 0 α1
...
...
. . .
...
...
. . .
...
...
0 0 · · · X + u αn−2 · · · αk αk−1
−1 X · · · 0 ck
−1
. . .
...
. . . X cn−2
−1 X + cn−1
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= (X + u)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
X + u 0 · · · 0 0 · · · 0 α1
−1 X + u · · · 0 0 · · · 0 α2
...
...
. . .
...
...
. . .
...
...
0 0 · · · X + u αn−2 · · · αk αk−1
−1 X · · · 0 ck
−1
. . .
...
. . . X cn−2
−1 X + cn−1
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
+ α0.
It is easy to find that the coefficient of Xn−1 in |XIn−M | is k ·u+ cn−1.
By division algorithm, we obtain f = (X+u)q+α0 and |XIn−M | = f
if and only if
q =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
X + u 0 · · · 0 0 · · · 0 α1
−1 X + u · · · 0 0 · · · 0 α2
...
...
. . .
...
...
. . .
...
...
0 0 · · · X + u αn−2 · · · αk αk−1
−1 X · · · 0 ck
−1
. . .
...
. . . X cn−1
−1 X + cn−1
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
.
Again, the coefficient of Xn−2 in q is (k − 1)·u + cn−1. We apply in-
duction hypothesis so that determine (uniquely) α0, α1, · · · , αn−2. This
completes our proof.
276 Nil-quasi-clean companion matrices
Lemma 2. Let A ∈ Mn(F ) and A = E +B be a nil-quasi-clean decom-
position, where E2 = uE and Bm = 0 for some positive integer m. Then
there exists a positive integer k such that trace(A) = k · u. Moreover,
(1) If char(F ) = 0 and trace(A) = k · u, then
(a) k ⩽ n;
(b) k = 0 if and only if A is nilpotent;
(c) k = n if and only if A = uIn +B, where B is nilpotent.
(2) If char(F ) = p > 0, then
(a) there exists k ∈ {1, 2, · · · , p} such that trace(A) = k · u, and
k ⩽ n or k = p;
(b) if n < k = p, then A is nilpotent;
(c) if k = n < p, then A = uIn +B, where B is nilpotent;
(d) if k = n = p, then A is nilpotent or A = uIn +B, where B is
nilpotent.
Proof. Since A = E + B, we have trace(A) = trace(E) + trace(B) =
trace(E). Because E2 = uE, we know that trace(E) = u · rank(E),
there is k ∈ N such that trace(A) = k · u. Note that k is unique if
char(F ) = 0 and k ⩽ n, so the statement (1)(a) is obvious. Note k is
unique only modulo p if char(F ) = p, hence (2)(a) is also hold.
(1)(b) If k = 0, then rank(E) = 0, so A is nilpotent. The converse
is obvious.
(1)(c) If k = n, then rank(E) = n, so A = uIn + B, where B is
nilpotent. The converse is obvious.
(2)(b) If k = p, then trace(A) = trace(E) = u ·rank(E) = 0, hence
p | rank(E). We know rank(E) ⩽ n < p, so rank(E) = 0, i.e., E = 0,
so A is nilpotent.
(2)(c) If k = n, then rank(E) ≡ n (mod p). Since rank(E) ⩽ n <
p, so rank(E) = n and it follows that A = uIn +B with B is nilpotent.
(2)(d) As A = E + B, where E2 = uE ∈ QId(Mn(F )) and B ∈
Nil(Mn(F )), we have trace(A) = u · rank(E) = 0, so rank(E) ∈ {0, p}.
This implies E ∈ {0, uIn}. This completes the proof.
Theorem 1. Let C = Cc0,c1,...,cn−1 be a companion matrix over a field F .
Then C is nil-quasi-clean if and only if one of the following conditions
holds:
H. Su, S. Liu 277
(1) C is nilpotent matrix;
(2) C = uIn + B with B is nilpotent matrix; (i.e. ci = (−u)i
(
n
n−i
)
for
all i ∈ {0, 1, · · · , n− 1});
(3) char(F ) = 0 and there exists a positive integer k such that −cn−1 =
k · u and n > k;
(4) char(F ) = p and there exists k ∈ {1, 2, · · · , p} such that −cn−1 =
k·u and n > k.
Proof. ⇒ Suppose that C = E + B with E2 = uE ∈ QId(Mn(F ))
and B ∈ Nil(Mn(F )). We may assume that C is not a nilpotent and
C ̸= uIn +B.
If char(F ) = 0, by Lemma 2(1), we have that there exists a unique
k ⩽ n such that −cn−1 = k·u. Since C is neither nilpotent nor uIn +B,
we have 0 < k < n.
If char(F ) = p, by Lemma 2(2), there is a unique k ∈ {1, 2, · · · , p}
such that −cn−1 = k · u. If 0 ̸= E ̸= uIn is a quasi-idempotent such
that C − E is nilpotent matrix, then rank(E) ≡ k (mod p). As 0 ̸=
rank(E) ̸= n, we have k ⩽ rank(E) < n and hence n > k.
⇐ If C is nilpotent matrix or C = uIn + B, then C is obviously
nil-quasi-clean. If we are in one of the cases (3) or (4), we can apply
Proposition 3 for g = 0 to obtain a nil-quasi-clean decomposition for C.
Theorem 2. Let n ⩾ 3 be an integer. The following statements are
equivalent for a field F :
(1) every companion matrix C ∈ Mn(F ) is nil-quasi-clean;
(2) char(F ) = p < n;
(3) if C ∈ Mn(F ) is a companion matrix then for every polynomial g ∈
F [X] of degree at most n−2 there exist two matrices E,M ∈ Mn(F )
such that C = E + M , E2 = uE with u ∈ F ∗ and |XIn − M | =
Xn + g.
Proof. (2)⇒(3) For every companion matrix C, we have trace(C) =
−cn−1 = k·u with k ∈ {1, 2, · · · , p} and we use Proposition 3 and the
result follows.
(3)⇒(1) It is obvious.
278 Nil-quasi-clean companion matrices
(1)⇒(2) Since every element of the field F can be the trace of a com-
panion matrix, by Lemma 2, every element from F has the form k · u,
k ∈ N. This implies that there exists a prime p such that char(F ) = p.
Moreover, suppose p ⩾ n, then use Theorem 1 to observe that the com-
panion matrix C =
0 0 0 · · · 0 0
1 0 0 · · · 0 −1
0 1 0 · · · 0 0
. . .
. . .
...
...
. . . 0 0
1 0
is not nil-quasi-clean.
Since trace(C) = 0, so p|rank(E) and rank(E) ⩽ n ⩽ p. Hence
rank(E) = 0, which means C is nilpotent, contradiction. Or rank(E) =
p = n, which means E = uIn and C − E is nilpotent, C − E =
−u 0 0 · · · 0 0
1 −u 0 · · · 0 −1
0 1 −u · · · 0 0
. . .
. . .
...
...
. . . −u 0
1 −u
is not nilpotent matrix, a contradic-
tion. Therefore, p < n, and the proof is complete.
Finally, we illustrate above theorem by an example of companion
matrix of order 4. Note that for n = 4, the characteristics of a field is
only 2 or 3.
Example 1. First, for n = 4 and char(F ) = 2:
(i) for c3 = 0 we have
C =
u 0 0 u4
1 u c2 − u2 uc2 − u3
0 1 0 −c2
0 0 1 0
+
−u 0 0 −u4 − c0
0 −u u2 − c2 u3 − c1 − uc2
0 0 0 0
0 0 0 0
= N + E, where N is nilpotent and E2 = uE for some u ∈ F ∗.
(ii) for c3 ̸= 0 we have
C =
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
+
0 0 0 −c0
0 0 0 −c1
0 0 0 −c2
0 0 0 −c3
= N + E,
H. Su, S. Liu 279
where N ∈ Nil(M4(F )) and E2 = −c3E ∈ QId(M4(F )).
Secondly, for n = 4 and char(F ) = 3:
(i) for c3 = 0 we have
C =
u 0 0 −u4
1 u 0 −u3
0 1 u 0
0 0 1 0
+
−u 0 0 u4 − c0
0 −u 0 u3 − c1
0 0 −u −c2
0 0 0 0
= N + E,
where N is nilpotent and E2 = −uE for some u ∈ F ∗.
(ii) for c3 ̸= 0 we have
C =
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
+
0 0 0 −c0
0 0 0 −c1
0 0 0 −c2
0 0 0 −c3
= N + E,
where N ∈ Nil(M4(F )) is nilpotent and E2 = −c3E ∈ QId(M4(F )).
References
[1] Al Habibi, M.F.M., Irawati, S., Susanto, H., Sulandra, I.M., Marubayashi, H.,
Ambarsari, I.F.: Sufficient condition for a nil-clean element to be clean in a
certain subring of M3(Z). Compusoft 8(9), 3410–3414 (2019).
[2] Abyzov, A.N., Mukhametgaliev, I.I.: On some matrix analogs of the little Fermat
theorem. Math. Notes 101(2), 187–192 (2017). https://doi.org/10.1134/S0001434
617010229
[3] Breaz, S., Modoi, G.C.: Nil-clean companion matrices. Linear Algebra Its Appl.
489, 50–60 (2016). https://doi.org/10.1016/j.laa.2015.10.005
[4] Breaz, S., Cǎlugǎreanu, G., Danchev., P., Micu, T.: Nil-clean matrix rings. Linear
Algebra Its Appl. 439(10), 3115–3119 (2013). https://doi.org/10.1016/j.laa.2013.
08.027.
[5] Cǎlugǎreanu, G.: The nil-clean 2 × 2 integral units. Hacettepe J. Math. Stat.
50(1), 41–45 (2021). https://doi.org/10.15672/hujms.622655
[6] Cǎlugǎreanu, G.: Some matrix completions over integral domains. Linear Algebra
Its Appl. 507, 414–419 (2016). https://doi.org/10.1016/j.laa.2016.06.034
[7] Diesl, A.J.: Nil clean rings. J. Algebra 383, 197–211 (2013). https://doi.org/10.10
16/j.jalgebra.2013.02.020
[8] Šter, J.: On expressing matrices over Z2 as the sum of an idempotent and a nilpo-
tent. Linear Algebra Its Appl. 544, 339–349 (2018). https://doi.org/10.1016/j.laa.
2018.01.015
https://doi.org/10.1134/S0001434617010229
https://doi.org/10.1134/S0001434617010229
https://doi.org/10.1016/j.laa.2015.10.005
https://doi.org/10.1016/j.laa.2013.08.027
https://doi.org/10.1016/j.laa.2013.08.027
https://doi.org/10.15672/hujms.622655
https://doi.org/10.1016/j.laa.2016.06.034
https://doi.org/10.1016/j.jalgebra.2013.02.020
https://doi.org/10.1016/j.jalgebra.2013.02.020
https://doi.org/10.1016/j.laa.2018.01.015
https://doi.org/10.1016/j.laa.2018.01.015
280 Nil-quasi-clean companion matrices
[9] Koşan, M. T., Lee., T.-K., Zhou, Y.: When is every matrix over a division ring
a sum of an idempotent and a nilpotent? Linear Algebra Its Appl. 450, 7–12
(2014). https://doi.org/10.1016/j.laa.2014.02.047
[10] Tang, G., Su, H., Yuan, P.: Quasi-clean rings and strongly quasi-clean rings.
Comm. Contemp. Math. 25(2), 2150079 (2023). https://doi.org/10.1142/S0219
199721500796
[11] Tang, G., Zhou, Y.: Nil G-cleanness and strongly nil G-cleanness of rings. J. Algeb-
ra Its Appl. 21(04), 2250077 (2022). https://doi.org/10.1142/S0219498822500773
Contact information
H. Su School of Science, Beibu Gulf University,
Qinzhou 535011, P.R. China
E-Mail: huadongsu@sohu.com
S. Liu College of Elementary Education, Chongzuo
Preschool Education College, Chongzuo
532200, P.R. China
E-Mail: shifeng4477@163.com
Received by the editors: 07.02.2026
and in final form 15.03.2026.
https://doi.org/10.1016/j.laa.2014.02.047
https://doi.org/10.1142/S0219199721500796
https://doi.org/10.1142/S0219199721500796
https://doi.org/10.1142/S0219498822500773
Huadong Su and Shifeng Liu
|
| id | admjournalluguniveduua-article-2466 |
| institution | Algebra and Discrete Mathematics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-09T01:00:17Z |
| publishDate | 2026 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| resource_txt_mv | admjournalluguniveduua/00/614fe73efc30b0be5681a6a969c82300.pdf |
| spelling | admjournalluguniveduua-article-24662026-07-08T07:55:33Z Nil-quasi-clean companion matrices Su, Huadong Liu, Shifeng quasi-idempotent matrix, nil-quasi-clean, nilpotent matrix, finite field 16U40, 15B33, 16S50 Let \(R\) be a ring with identity. An element \(e\) in \(R\) is called a quasi-idempotent element if \(e^2=ke\) for some central unit \(k\) in \(R\). For an element \(b\) in \(R\), if there is a positive integer \(m\) such that \(b^m=0\), then \(b\) is called a nilpotent element of \(R\). An element \(r\) in \(R\) is called a nil-quasi-clean element if \(r\) is a sum of a quasi-idempotent and a nilpotent. If every element of \(R\) is nil-quasi-clean, then \(R\) is called a nil-quasi-clean ring. This paper completely determines nil-quasi-clean companion matrices over a field. Lugansk National Taras Shevchenko University 2026-07-08 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2466 10.12958/adm2466 Algebra and Discrete Mathematics; Vol 41, No 2 (2026) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2466/pdf Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | quasi-idempotent matrix nil-quasi-clean nilpotent matrix finite field 16U40 15B33 16S50 Su, Huadong Liu, Shifeng Nil-quasi-clean companion matrices |
| title | Nil-quasi-clean companion matrices |
| title_full | Nil-quasi-clean companion matrices |
| title_fullStr | Nil-quasi-clean companion matrices |
| title_full_unstemmed | Nil-quasi-clean companion matrices |
| title_short | Nil-quasi-clean companion matrices |
| title_sort | nil-quasi-clean companion matrices |
| topic | quasi-idempotent matrix nil-quasi-clean nilpotent matrix finite field 16U40 15B33 16S50 |
| topic_facet | quasi-idempotent matrix nil-quasi-clean nilpotent matrix finite field 16U40 15B33 16S50 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2466 |
| work_keys_str_mv | AT suhuadong nilquasicleancompanionmatrices AT liushifeng nilquasicleancompanionmatrices |