An application of the Zhou radical to the \(e\)-reversibility of rings
Let \(R\) be a ring and \(e\) be an idempotent element of \(R\). The Zhou radical of \(R\) denoted by \(\delta(R)\) is the intersection of maximal essential right ideals of \(R\). In the literature, \(e\)-reversible rings were studied regarding the question of how idempotent elements affect the reve...
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Algebra and Discrete Mathematics| _version_ | 1870196964052697088 |
|---|---|
| author | Ungor, Burcu Kose, Handan Harmanci, Abdullah |
| author_facet | Ungor, Burcu Kose, Handan Harmanci, Abdullah |
| author_institution_txt_mv | [
{
"author": "Burcu Ungor",
"institution": "Ankara University"
},
{
"author": "Handan Kose",
"institution": "Ankara University"
},
{
"author": "Abdullah Harmanci",
"institution": "Hacettepe University"
}
] |
| author_sort | Ungor, Burcu |
| 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 and \(e\) be an idempotent element of \(R\). The Zhou radical of \(R\) denoted by \(\delta(R)\) is the intersection of maximal essential right ideals of \(R\). In the literature, \(e\)-reversible rings were studied regarding the question of how idempotent elements affect the reversible property of rings. In this paper, we provide an application of the Zhou radical of a ring to the reversibility depending on idempotents. Accordingly, we study rings \(R\) in which \(ab = 0\) implies \(bae\in\delta(R)\) (or \(eba\in\delta(R),\)) where \(a, b\in R\), called Zhou right (or left) \(e\)-reversible rings. Besides studying the structure of Zhou \(e\)-reversible rings, we investigate relations between Zhou \(e\)-reversible rings and some known rings, such as Zhou \(e\)-reduced rings, central reversible rings, NI-rings, semiperfect rings and matrix rings. In addition to these, we determine the Zhou radical of some certain rings, such as Morita context rings and special subrings of a direct product of rings. |
| doi_str_mv | 10.12958/adm2462 |
| first_indexed | 2026-07-09T01:00:11Z |
| format | Article |
| fulltext |
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 41 (2026). Number 2, pp. 281–304
DOI:10.12958/adm2462
An application of the Zhou radical to the
e-reversibility of rings
Burcu Ungor, Handan Kose, and Abdullah Harmanci
Communicated by L. Kurdachenko
Abstract. Let R be a ring and e be an idempotent element
of R. The Zhou radical of R denoted by δ(R) is the intersection of
maximal essential right ideals of R. In the literature, e-reversible
rings were studied regarding the question of how idempotent ele-
ments affect the reversible property of rings. In this paper, we pro-
vide an application of the Zhou radical of a ring to the reversibility
depending on idempotents. Accordingly, we study rings R in which
ab = 0 implies bae ∈ δ(R) (or eba ∈ δ(R)), where a, b ∈ R, called
Zhou right (or left) e-reversible rings. Besides studying the structu-
re of Zhou e-reversible rings, we investigate relations between Zhou
e-reversible rings and some known rings, such as Zhou e-reduced
rings, central reversible rings, NI-rings, semiperfect rings and mat-
rix rings. In addition to these, we determine the Zhou radical of
some certain rings, such as Morita context rings and special sub-
rings of a direct product of rings.
Introduction
Throughout this paper, all rings are associative with identity. For a ring
R, we use N(R), J(R), U(R), C(R) and Id(R) to denote the set of all
nilpotent elements, the Jacobson radical, the set of all invertible elements,
the center and the set of all idempotent elements of R, respectively. Also,
2020 Mathematics Subject Classification: 16U40, 16U80, 16N40, 16U99,
16S50.
Key words and phrases: reversible ring, e-reversible ring, Zhou radical, idem-
potent element, Morita context.
https://doi.org/10.12958/adm2462
282 An application of the Zhou radical
δ(R) stands for the intersection of maximal essential right ideals of a
ring R. The ideal δ(R) is defined in [20] and named as Zhou radical in [5].
The n×n full (resp., upper triangular) matrix ring over R is denoted by
Mn(R) (resp., Un(R)), and Dn(R) denotes the subring of Un(R) having
all diagonal entries are equal, and Vn(R) = {(aij) ∈ Dn(R) | aij =
a(i+1)(j+1) for i = 1, . . . , n−2 and j = 2, . . . , n−1} is a subring of Dn(R).
The ring of integers and the ring of integers modulo n are denoted by Z
and Zn, respectively.
The notion of reduced ring and its various generalizations have been
comprehensively studied in the literature. A ring is called reduced if it has
no nonzero nilpotent elements. Reduced rings are extended to e-reduced
rings in [14]. Let R be a ring and e ∈ Id(R). Then R is called left (or
right) e-reduced if eN(R) = 0 (or N(R)e = 0). As a generalization of the
notion of e-reduced ring, in [11], a ring R is said to be Zhou right (resp.,
left) e-reduced provided that N(R)e ⊆ δ(R) (resp., eN(R) ⊆ δ(R)). A
ring R is called Zhou e-reduced if it is both Zhou right e-reduced and
Zhou left e-reduced.
Reversible rings, as a natural common generalization of commutative
rings, integral domains and reduced rings, were studied by Cohn in [1].
There are many papers to investigate reversible rings and their genera-
lizations. For instance, in [7], as a generalization of reversible rings,
central reversible rings were investigated. A ring R is called central re-
versible if for any a, b ∈ R, having ab = 0 implies that ba is central in R.
A version of reversibility depending on idempotents was studied in [9] as
another generalization of reversible rings. In this direction, a ring R is
said to be right (resp., left) e-reversible if for any a, b ∈ R, having ab = 0
implies bae = 0 (resp., eba = 0). The ring R is called e-reversible if it
is both left and right e-reversible. In [10], e-reversibility of rings was
discussed from the perspective of quasinilpotents as a generalization of
e-reversible rings.
In ring theory, the Zhou radical, idempotent elements, the reversibili-
ty and related notions have important roles and generated wide interest.
With this motivation, in this paper, we relate these concepts and give an
application of the Zhou radical to the e-reversibility of rings. In this per-
spective, we consider “Zhou e-reversibility”. The contents of the paper
is as follows: In Section 2, we deal with the Zhou radical of certain rings
to use in the sequel of the paper. Within this scope, we determine the
Zhou radicals of the Morita context and the ring R[A,B] which is a sub-
ring of a direct product of copies of A, where A is a ring and B is a sub-
B. Ungor, H. Kose, A. Harmanci 283
ring of A. In Section 3, we concentrate on the Zhou e-reversible rings,
and exhibit some sources for Zhou e-reversible rings. Also, some results
related to the structure of Zhou right e-reversible rings are observed.
In Section 4, we focus on some ring extensions in terms of the Zhou
e-reversibility. Finally, in Section 5, certain matrix rings are investigated
related to the Zhou e-reversible property.
1. Notes on the Zhou radical
As a generalization of small submodules, δ-small submodules introduced
by Zhou in [20] to study δ-semiperfect modules. Let M be a module
and N a submodule of M . Then N is called δ-small in M if whenever
M = N +L and M/L is singular, then M = L. The sum of δ-small sub-
modules is denoted by δ(M). Considering the ring R as a right R-module
over itself, the ideal δ(R) is introduced as the sum of δ-small right ideals
of R. We begin with some properties of δ(R) that will be used in the
sequel.
Let Sr denote the right socle of the ring R, that is, Sr is the sum of
minimal right ideals of R. Then J(R/Sr)=δ(R)/Sr by [20, Corollary1.7].
It is clear by definitions that J(R) ⊆ δ(R). The next example shows
that this inclusion is strict. Also, in case N(R) is an ideal of R, we have
N(R) ⊆ δ(R).
Examples 1.1. (1) Consider the ring R = U2(Z2). Then
J(R) =
[
0 Z2
0 0
]
and δ(R) =
[
0 Z2
0 Z2
]
.
This shows J(R) ⫋ δ(R).
(2) Let R =M2(Z). Then J(R) = δ(R) = 0.
The following properties relating to the Zhou radical are obvious.
Lemma 1.2. Let R be a ring and n a positive integer.
(1) Let Ii denote the ith-row of Un(R). Then δ(Un(R)) =
n∑
i=1
δ(Ii).
(2) δ(Dn(R)) = {(aij) ∈ Dn(R) | aii ∈ δ(R)}.
(3) δ(Vn(R)) = {(aij) ∈ Vn(R) | aii ∈ δ(R)}.
(4) δ(Mn(R)) =Mn(δ(R)).
284 An application of the Zhou radical
A Morita context is a 6-tuple M = (R, V,W, S, ϕ, ψ), where R,S are
rings, RVS and SWR are bimodules with context products ϕ : V ×W → R
and ψ : W×V → S written multiplicatively as (v, w) 7→ vw and (w, v) 7→
wv such that T (M) =
[
R V
W S
]
is an associative ring with the obvious
matrix operations. The ring T (M) is the Morita context ring associated
with M. The ring T (M) is called trivial if the context products are
trivial, i.e., VW = 0 and WV = 0 (see for detail [16] and [19]). This is
also called null context. In [18], maximal ideals of T (M) are determined
as follows.
Lemma 1.3 ([18]). Let M = (R, V,W, S) be a Morita context and
T (M) =
[
R V
W S
]
be the Morita context ring. Then the following hold.
(1) Let I be a maximal right ideal of R and VI = {v ∈ V | vW ⊆ I}
and MI =
[
I VI
W S
]
. Then VI is a right S-submodule of V and MI
is a maximal right ideal of T (M).
(2) Let J be a maximal right ideal of S and WJ = {w ∈W | wV ⊆ J}
and MJ =
[
R V
WJ J
]
. Then WJ is a right R-submodule of W and
MJ is a maximal right ideal of T (M).
(3) Let K be a maximal right ideal of T (M) which is a different type
from ones in (1) and (2). Then
[
J(R) V0
W0 J(S)
]
⊆ K, where V0 =
{v ∈ V | vW ⊆ J(R)} and W0 = {w ∈W | wV ⊆ J(S)}.
In the following, we investigate the Zhou radical of Morita context
rings. First, we start with the essential maximal right ideals of Morita
context rings.
Lemma 1.4. Let M = (R, V,W, S) be a Morita context with the Morita
context ring T (M). Let I be an essential maximal right ideal of R and
J be an essential maximal right ideal of S. Then the following hold.
(1) Let VI = {v ∈ V | vW ⊆ I} and MI =
[
I VI
W S
]
. Then VI is a
right S-submodule of V and MI is an essential maximal right ideal
of T (M).
B. Ungor, H. Kose, A. Harmanci 285
(2) Let WJ = {w ∈ W | wV ⊆ J} and MJ =
[
R V
WJ J
]
. Then WJ is
a right R-submodule of W and MJ is an essential maximal right
ideal of T (M).
Proof. The maximality of MI and MJ are known by Lemma 1.3. As for
MI being an essential right ideal of T (M), we assume otherwise. So there
exists a right ideal K of T (M) such that MI ⊕K = T (M). Multiplying
from both sides by e =
[
1 0
0 0
]
, we have eMIe⊕ eKe = eT (M)e. Hence
I ⊕ eKe = R. It entails that I is not essential. This is a contradiction.
Thus MI is an essential right ideal of T (M). On the other hand, MJ
being an essential right ideal of T (M) is treated similarly.
Theorem 1.5. Let M = (R, V,W, S) be a Morita context with the
Morita context ring T (M). Then δ(T (M)) ⊆
[
δ(R) V1
W1 δ(S)
]
, where
V1 = {v ∈ V | vW ⊆ δ(R)} and W1 = {w ∈W | wV ⊆ δ(S)}.
Proof. Since δ(T (M)) is an ideal of T (M), by [19, Lemma 2.1(3)],
δ(T (M)) =
[
A K
L B
]
, where A is an ideal of R, B is an ideal of S,
K is a submodule of RVS and L is a submodule of SWR with KW ⊆ A,
WK ⊆ B, LV ⊆ B, V L ⊆ A, AV ⊆ K, WA ⊆ L, BW ⊆ L and
V B ⊆ K. We claim that A ⊆ δ(R) and B ⊆ δ(S). Assume contrary
that A ̸⊆ δ(R). So there exists an essential maximal right ideal I of R
such that A ̸⊆ I. Consider the right ideal MI =
[
I VI
W S
]
of T (M),
where VI = {v ∈ V | vW ⊆ I}. By Lemma 1.4(1), MI is an essen-
tial maximal right ideal of T (M). It entails δ(T (M)) ⊆ MI , and so
A ⊆ I, a contradiction. Hence A ⊆ δ(R). By a similar discussion, we
obtain B ⊆ δ(S). Consider the sets V1 = {v ∈ V | vW ⊆ δ(R)} and
W1 = {w ∈ W | wV ⊆ δ(S)}. Having KW ⊆ A and LV ⊆ B yield
K ⊆ V1 and L ⊆W1, respectively. It follows
δ(T (M)) =
[
A K
L B
]
⊆
[
δ(R) V1
W1 δ(S)
]
.
Theorem 1.6. Let M = (R, V,W, S) be a trivial Morita context with
the Morita context ring T (M). Then δ(T (M)) =
[
δ(R) V
W δ(S)
]
.
286 An application of the Zhou radical
Proof. In a trivial Morita context, V1 = V and W1 = W , and so
δ(T (M)) ⊆
[
δ(R) V
W δ(S)
]
by Theorem 1.5. For the reverse inclusion,
consider the right ideals X =
[
0 V
0 0
]
and Y =
[
0 0
W 0
]
of T (M).
Note that X and Y are nilpotent right ideals. It follows that X,Y ⊆
δ(T (M)). Hence X + Y ⊆ δ(T (M)). By [20, Theorem 1.6(2) and
Lemma 1.3(1)], X+Y is δ-small in T (M). By the fact that δ(R) and δ(S)
are δ-small in R and S, respectively,
[
δ(R) V
W δ(S)
]
/(X+Y ) is δ-small in[
R V
W S
]
/(X + Y ), and so
[
δ(R) V
W δ(S)
]
is δ-small in
[
R V
W S
]
by [20,
Lemma 1.3(1)]. Thus
[
δ(R) V
W δ(S)
]
⊆ δ(T (M)). Therefore δ(T (M)) =[
δ(R) V
W δ(S)
]
.
We illustrate Theorem 1.6 by the following example.
Example 1.7. Let M denote the Morita context M=(Z6, 2̄Z6, 3̄Z6,Z6)
and T (M) be a Morita context ring T (M) =
[
Z6 2̄Z6
3̄Z6 Z6
]
. Since M is a
trivial Morita context, we have δ(T (M))=
[
δ(Z6) 2̄Z6
3̄Z6 δ(Z6)
]
by Theorem1.6.
The ring Z6 being semisimple entails that δ(Z6) = Z6. It follows δ(T (M))
= T (M).
In the next result, we investigate under what conditions the reverse
inclusion in Theorem 1.5 holds.
Theorem 1.8. Let M = (R, V,W, S) be a Morita context with the
Morita context ring T (M). If δ(R) and δ(S) are nil ideals of R and S,
respectively, then δ(T (M)) =
[
δ(R) V1
W1 δ(S)
]
, where V1 = {v ∈ V | vW ⊆
δ(R)} and W1 = {w ∈W | wV ⊆ δ(S)}.
Proof. By Theorem 1.5, δ(T (M)) ⊆
[
δ(R) V1
W1 δ(S)
]
. For the reverse
inclusion, consider the sets X =
[
δ(R) V1
0 0
]
and Y =
[
0 0
W1 δ(S)
]
.
On the one hand, X and Y are right ideals of T (M). On the other
hand, since δ(R) and δ(S) are nil ideals, X and Y are also nil. It
B. Ungor, H. Kose, A. Harmanci 287
follows that X,Y ⊆ δ(T (M)). Hence X + Y ⊆ δ(T (M)), that is,[
δ(R) V1
W1 δ(S)
]
⊆ δ(T (M)). This completes the proof.
Lemma 1.9. Let M = (R, V,W, S) be a Morita context with the Morita
context ring T (M) and A =
[
r v
w s
]
. Then the following hold.
(1) If A ∈ Id(T (M)), then r ∈ Id(R) and s ∈ Id(S).
(2) If A ∈ N(T (M)), then r ∈ N(R) and s ∈ N(S).
Proof. Clear by definitions.
In [2], the Dorroh extension of a ring R was introduced by Dorroh
as a way to embed a ring R without an identity into a ring with an
identity Z ⊕ R, and it is one of the important methods of constructing
new rings and analyzing some properties of rings. Let R be a ring and
S be an associative ring that may not possess an identity element and
an (R,R)-bimodule obeying multiplication in S, that is, for any a ∈ R
and s, t ∈ S, a(ts) = (at)s, t(as) = (ta)s and (ts)a = t(sa). The
Dorroh extension (in other words, ideal extension) of S by R, denoted
by D(R,S), is the abelian group R × S with multiplication defined by
(a1, t1)(a2, t2) = (a1a2, a1t2 + t1a2 + t1t2) for a1, a2 ∈ R and t1, t2 ∈ S.
Then (1, 0) is the identity of D(R,S). Maximal ideals and right (or
left) ideals of Dorroh extensions were characterized by Mesyan in [15,
Proposition 5]. The Zhou radical δ(D(R,S)), idempotents and nilpotents
of D(R,S) are characterized in [11, Lemma 2.8] as the following.
Lemma 1.10. Let S be an algebra over a ring R and consider the Dorroh
extension D(R,S) of S by R. Let (r, s) ∈ D(R,S). Then we have the
following.
(1) δ(D(R,S)) = δ(R)⊕ S.
(2) (r, s) ∈ Id(D(R,S)) if and only if r ∈ Id(R) and (r + s)2 = r + s.
(3) (r, s) ∈ N(D(R,S)) with (r, s)n = 0 if and only if rn = 0 and
(r + s)n = 0.
Let A be a ring and B a subring of A and R[A,B] denote the set
R[A,B] = {(a1, a2, . . . , an, b, b, . . . ) | ai ∈ A, b ∈ B, 1 ≤ i ≤ n, n ∈ Z+}.
288 An application of the Zhou radical
Then R[A,B] is a ring under the componentwise addition and multiplica-
tion. In the following, we determine the right socle and the Zhou radical
of this ring.
Theorem 1.11. Let A be a ring and B a subring of A. Then
Soc(R[A,B]) = R[Soc(A), 0].
Proof. Let I be a minimal right ideal of A. We claim that R[I, 0] is a
minimal right ideal of R[A,B]. It is clear that R[I, 0] is a right ideal
of R[A,B]. For the minimality, let Y be a right ideal of R[A,B] with
Y ⊆ R[I, 0]. Set
IY = {a ∈ A |there exists (a1, a2, . . . , an, 0, 0, . . . ) ∈ Y,
for some i, a = ai}.
Let < IY > denote the right ideal generated by IY . Then < IY >⊆ I and
R[< IY >, 0] ⊆ Y ⊆ R[I, 0]. Since I is minimal, I =< IY >. Thus
Y = R[I, 0], i.e., R[I, 0] is minimal. So R[Soc(A), 0] ⊆ Soc(R[A,B]).
Let X be a minimal right ideal of R[A,B] and
x = (a1, a2, . . . , an, b, b, . . . ) ∈ X.
Multiplying x from the right by t = (1, 1, . . . , 1︸ ︷︷ ︸
n times
, 0, 0, . . . ) ∈ R[A,B], we
get xt = (a1, a2, . . . , an, 0, 0, . . . ) ∈ X. Set
S = {(a1, a2, . . . , an, 0, 0, . . . ) ∈ R[A,B] | there exists
(a1, a2, . . . , an, b, b, . . . ) ∈ X}.
Then S ⊆ X ⊆ R[A,B]. Since S is a right ideal of R[A,B] and X is
simple, S = X. We now say that Soc(R[A,B]) ⊆ R[A, 0].
Let IS denote the right ideal of A generated by the entries of the
elements of S. We may conclude that R[Is, 0] = S since S is minimal.
Next we claim that IS is a minimal right ideal of A. Let J be a right ideal
of A with J ⊆ Is. Then R[J, 0] ⊆ R[Is, 0]. Since R[J, 0] is a right ideal
and R[Is, 0] is a minimal right ideal, R[J, 0] = R[Is, 0]. Thus J = IS .
It entails that Soc(R[A,B]) ⊆ R[Soc(A), 0]. Therefore Soc(R[A,B]) =
R[Soc(A), 0].
Theorem 1.12. Let A be a ring and B a subring of A. Then the fol-
lowing hold:
(1) J(R[A,B]) = R[J(A), J(A) ∩ J(B)];
B. Ungor, H. Kose, A. Harmanci 289
(2) U(R[A,B]) = R[U(A), U(B)];
(3) δ(R[A,B]) = R[δ(A), δ(A) ∩ δ(B)].
Proof. (1) It is proved in [4, Lemma 3.11].
(2) Let X = (a1, a2, . . . , an, b, b, . . . ) ∈ U(R[A,B]), and let Y =
(y1, y2, . . . , yn, x, x, . . . ) ∈ U(R[A,B]) be with XY = Y X = (1, 1, 1, . . . ).
Then aiyi = yiai = 1 (i = 1, 2, 3, . . . , n), bx = xb = 1. Hence ai ∈ U(A),
b ∈ U(B), and so X ∈ R[U(A), U(B)]. Let X = (a1, a2, . . . , an, b, b, . . . )
∈ R[U(A), U(B)]). Then a−1
i ∈ U(A) (i = 1, 2, 3, . . . , n), b−1 ∈ U(B).
Write X−1 = (a−1
1 , a−1
2 , . . . , a−1
n , b−1, b−1, b−1, . . . ) ∈ R[A,B]. Then
XX−1 = X−1X = 1 ∈ R[A,B]. Hence X ∈ U(R[A,B]).
(3) It is known that
J(R[A,B]/Soc(R[A,B])) = δ(R[A,B])/Soc(R[A,B]).
By (1) and Theorem 1.11, we have
J(R[A,B]/Soc(R[A,B]))
= J(R[A,B]/R[Soc(A), 0])
= J(R[A/Soc(A), (B + Soc(A))/Soc(A)])
= R[J(A/Soc(A)), J(A/Soc(A)) ∩ J((B + Soc(A))/Soc(A))]
= R[δ(A)/Soc(A), (δ(A)/Soc(A)) ∩ (δ(B + Soc(A))/Soc(A))]
= R[δ(A)/Soc(A), (δ(A)/Soc(A)) ∩ ((δ(B) + Soc(A))/Soc(A))]
= R[δ(A)/Soc(A), (δ(A) ∩ (δ(B) + Soc(A)))/Soc(A)]
= R[δ(A)/Soc(A), ((δ(A) ∩ δ(B)) + Soc(A))/Soc(A)].
It follows that δ(R[A,B]) = R[δ(A), δ(A) ∩ δ(B)].
2. Application of the Zhou radical to the e-reversibility
In this section, we present an application of the Zhou radical to the
e-reversibility of rings. In this direction, we give our main definition as
follows.
Definition 2.1. Let R be a ring, e ∈ Id(R) and a ∈ R. Then a
is called Zhou right e-reversible if whenever ab = 0 for any b ∈ R,
bae ∈ δ(R). The ring R is called Zhou right e-reversible if every ele-
ment of R is Zhou right e-reversible. Zhou left e-reversible rings are
defined similarly, i.e., having ab = 0 implies eba ∈ δ(R). The ring R
is said to be Zhou e-reversible if every element of R is both Zhou right
e-reversible and Zhou left e-reversible.
290 An application of the Zhou radical
In the following, we study the Zhou e-reversibility as an element-wise
notion, and present an example to exhibit that the Zhou e-reversibility
of elements in rings is not left-right symmetric. In all of the cases in
the context, however, we discuss Zhou e-reversibility only as a right side
condition.
Example 2.2. Consider the ring R =M2(Z), A =
[
0 1
0 1
]
∈ R and E =[
1 1
0 0
]
∈ Id(R). Let B =
[
a b
c d
]
∈ R with AB = 0. Then c = d = 0, and
so BAE = 0 ∈ δ(R). Hence A is Zhou right E-reversible. On the other
hand, let C =
[
1 1
0 0
]
∈ R. Thus AC = 0, but ECA =
[
0 2
0 0
]
/∈ δ(R).
Therefore A is not Zhou left E-reversible.
It is clear that every ring is Zhou 0-reversible. Also, a ring R is Zhou
right (equivalently, left) 1-reversible, then R is Zhou e-reversible for every
e ∈ Id(R). In the sequel, we assume that e ∈ Id(R)\{0}. Obviously, every
right e-reversible ring is Zhou right e-reversible. Also, if N(R) ⊆ δ(R) for
a ring R, then R is Zhou e-reversible for every e ∈ Id(R). We illustrate
the notion of Zhou e-reversible rings in the following.
Example 2.3. Let R = A/I denote the ring in [11, Example 2.4] defined
by the ring A = Z2 < a, b > be the free algebra with noncommuting
indeterminates a, b over Z2, and I stand for the ideal generated by aAb,
a2 − a and b2 − b. We identify the elements in A with their images in R
for simplicity. Then
R = {0, 1, a, b, ba, a+ b, a+ ba, b+ ba, a+ b+ ba, 1 + a, 1 + b, 1 + ba, 1 +
a+ b, 1 + a+ ba, 1 + b+ ba, 1 + a+ b+ ba}
and aR = {0, a}, (ba)R = {0, ba}, (1+a+b+ba)R = {0, 1+a+b+ba} are
minimal right ideals of R. Hence Soc(RR) = aR⊕(ba)R⊕(1+a+b+ba)R
and δ(R) = {0, a, ba, a + ba, 1 + b, 1 + b + ba, 1 + b + a, 1 + b + a + ba}.
Also, Id(R) = {0, 1, a, b, 1 + a, 1 + b, b+ ba, 1 + a+ ba, a+ b+ ba, 1 + a+
b+ ba, a+ ba, 1+ b+ ba} and N(R) = {0, ba}. Thus having N(R) ⊆ δ(R)
yields that R is Zhou e-reversible for every e ∈ Id(R). On the other
hand, δ(U2(R)) =
[
δ(R) R
0 δ(R)
]
. Since N(U2(R)) ⊆ δ(U2(R)), U2(R) is
Zhou E-reversible for each E ∈ Id(U2(R)).
In [8], a ring R is called right (resp., left) e-semicommutative if for
any a, b ∈ R, having ab = 0 implies aRbe = 0 (resp., eaRb = 0).
B. Ungor, H. Kose, A. Harmanci 291
The ring R is called e-semicommutative in case R is both right and left
e-semicommutative.
Examples 2.4. The following are some sources for Zhou e-reversible
rings.
(1) Every semisimple ring is Zhou e-reversible.
(2) Every local ring is Zhou e-reversible.
(3) Every Zhou e-reduced ring is Zhou e-reversible.
(4) Every e-semicommutative ring is Zhou e-reversible.
(5) Every NI-ring is Zhou e-reversible.
(6) Every central reversible ring is Zhou e-reversible.
Proof. We only prove the statements for the right case, a similar proof
applies to the left case as well.
(1) Let R be a semisimple ring. Then δ(R) = R and so R is Zhou
e-reversible for each e ∈ Id(R).
(2) Let R be a local ring and a, b ∈ R with ab = 0. Then a ∈ J(R) or
b ∈ J(R). Hence bae ∈ J(R) ⊆ δ(R) for each e ∈ Id(R) in either case.
(3) Let R be a Zhou e-reduced ring with e ∈ Id(R). For any a, b ∈ R
such that ab = 0, we have ba ∈ N(R). Therefore bae ∈ δ(R).
(4) Assume that R is an e-semicommutative ring with e ∈ Id(R). Let
a, b ∈ R with ab = 0. Then (ba)(ba) = 0. Hence baRbae = 0. Having
baeR ⊆ baR yields (bae)R(bae) = 0. Thus R(bae) is a nilpotent left ideal
of R. It entails that R(bae) ⊆ δ(R) or bae ∈ δ(R).
(5) Assume that R is an NI-ring, i.e., N(R) is an ideal of R. Then
N(R) ⊆ J(R) ⊆ δ(R). Hence R is Zhou e-reversible for each e ∈ Id(R).
(6) It is a consequence of [7, Theorem 2.19] and (6).
Recall that a ring R is called right (quasi-)duo if every (maximal)
right ideal of R is two-sided. The next result gives another source for
Zhou e-reversible rings.
Theorem 2.5. Every right quasi-duo ring is Zhou right e-reversible for
every idempotent e.
Proof. Let R be a right quasi-duo ring and a, b ∈ R such that ab = 0.
Assume that bae /∈ δ(R) and we get a contradiction. There exists an
essential maximal right ideal I of R such that bae /∈ I. By hypothesis,
292 An application of the Zhou radical
I is an ideal and baeR + I = R. There exist r ∈ R and s ∈ I such that
baer+s = 1. Multiplying the latter from the left by a, we get as = a ∈ I.
Since I is an ideal, bae ∈ I. This is a contradiction. Thus bae ∈ δ(R).
Therefore R is Zhou right e-reversible.
In the sequel of this section, we observe some results related to the
structure of Zhou right e-reversible rings. In a ring R, any e ∈ Id(R) is
called right (resp., left) semicentral if er = ere (resp., re = ere) for all
r ∈ R.
Proposition 2.6. If R is a Zhou right e-reversible ring, then e = e +
δ(R) is a left semicentral idempotent in R/δ(R). The converse holds if
e(R/δ(R))e is reversible.
Proof. Since R is Zhou right e-reversible, having e(1 − e) = 0 yields
(1 − e)Re ⊆ δ(R). Therefore re − ere ∈ δ(R) for each r ∈ R, i.e.,
e ∈ R/δ(R) is left semicentral. For the converse statement, assume
that e(R/δ(R))e is reversible. Then R/δ(R) is right e-reversible by
[9, Proposition 2.9]. If a, b ∈ R such that ab = 0, then having ab = 0
yields bae = 0. This implies that bae ∈ δ(R).
We now give a characterization of Zhou right e-reversibility in terms
of nilpotents with nilpotency index 2.
Theorem 2.7. The following are equivalent for a ring R.
(1) R is Zhou right e-reversible.
(2) For any a ∈ R, if whenever a2 = 0, then ae ∈ δ(R).
Proof. (1) ⇒ (2) Let a ∈ R with a2 = 0. Assume that ae /∈ δ(R) and we
reach a contradiction. Let M be an essential maximal right ideal of M
such that ae /∈ M . Then a /∈ M . Hence there exist r ∈ R and m ∈ M
such that ar+m = 1. Multiplying the latter equality from the left by a,
we get a(1 −m) = 0. By (1), (1 −m)ae ∈ δ(R). Since (1 −m)ae ∈ M
and mae ∈M , we have ae ∈M . This contradicts the assumption. Thus
ae ∈ δ(R).
(2) ⇒ (1) Let a, b ∈ R with ab = 0. So (ba)2 = 0. By (2), bae ∈ δ(R).
To illustrate Theorem 2.7, we see the following examples.
Examples 2.8. (1) Let R be a reduced ring and n an integer with n ≥ 2.
Then Un(R) is Zhou right E-reversible for each E ∈ Id(Un(R)). Indeed,
B. Ungor, H. Kose, A. Harmanci 293
let A = (aij), B = (bij) ∈ Un(R) with AB = 0. Then aiibii = 0. By
assumption, biiaii = 0. Then the diagonal entries of BAE are zero for
any E ∈ Id(Un(R)). Since N(Un(R)) = {(aij) ∈ Un(R) | aii = 0}, we
have N(Un(R)) ⊆ δ(Un(R)). Therefore Un(R) is Zhou right E-reversible
for each E ∈ Id(Un(R)).
(2) Let F be a field. Then Mn(F ) is Zhou right E-reversible for each
E ∈ Id(Mn(F )) by the fact that δ(Mn(F )) =Mn(F ).
(3) The ring M2(Z) is not Zhou right E-reversible for E =
[
1 0
1 0
]
.
Indeed, consider A =
[
1 1
−1 −1
]
∈ N(M2(Z)). Then A2 = 0. But
AE =
[
2 0
−2 0
]
/∈ δ(M2(Z)) since δ(M2(Z)) = 0. Hence by Theorem 2.7,
M2(Z) is not Zhou right E-reversible.
Proposition 2.9. Let R be a Zhou right e-reversible ring. Then ab = 0
implies beae, aeb ∈ δ(R) for any a, b ∈ R.
Proof. Let a, b ∈ R such that ab = 0. Since R is Zhou right e-reversible,
bae ∈ δ(R). By Proposition 2.6, ae− eae ∈ δ(R). So bae− beae ∈ δ(R).
It follows beae ∈ δ(R). On the other hand, ab = 0 implies abr = 0
and so brae ∈ δ(R). Hence bRae ⊆ δ(R). Thus (aeb)R(aeb) ⊆ δ(R).
Since the Zhou radical is a semiprime ideal by [11, Proposition 2.6.],
aeb ∈ δ(R).
In what follows, we consider a condition (∗) under which bea ∈ δ(R)
in a Zhou right e-reversible ring R. This condition is compared to Propo-
sition 2.9.
For any a, b ∈ R, having ab = 0 implies bea ∈ δ(R) (∗)
Note that all reversible rings satisfy the condition (∗).
Theorem 2.10. Let R be a Zhou right e-reversible ring. Then the fol-
lowing are equivalent:
(1) R satisfies the (∗) condition;
(2) ex− exe ∈ δ(R) for any x ∈ R;
(3) e is central in R/δ(R).
294 An application of the Zhou radical
Proof. (1) ⇒ (2) Let x ∈ R and a = ex− exe, b = e. Then ab = 0. Since
R satisfies the (∗) condition, bea ∈ δ(R). But bea = ex − exe. Thus
ex− exe ∈ δ(R).
(2) ⇒ (1) Let a, b ∈ R with ab = 0. By hypothesis, ea − eae ∈ δ(R).
Multiplying ea−eae ∈ δ(R) from the left by b and use the fact that δ(R)
is an ideal of R, we have bea − beae ∈ δ(R). On the other hand, Zhou
right e-reversibility of R entails that beae ∈ δ(R) from Proposition 2.9.
Combining bea− beae ∈ δ(R) with beae ∈ δ(R), we conclude that bea ∈
δ(R).
(2) ⇒ (3) Note that e ∈ Id(R/δ(R)). Let x ∈ R. On the one hand, we
have ex − exe ∈ δ(R) by (2). On the other hand, xe − exe ∈ δ(R) by
Proposition 2.6. It follows that ex − xe ∈ δ(R). It entails ex = xe, as
desired.
(3) ⇒ (2) Let x ∈ R. Then ex = xe, and so ex−xe ∈ δ(R) by (3). Hence
(ex−exe)−(xe−exe) ∈ δ(R). Since xe−exe ∈ δ(R) by Proposition 2.6,
we have ex− exe ∈ δ(R).
The condition “being reduced of the ring” is not superfluous in Exam-
ples 2.8(1) by the following example.
Example 2.11. Let D be a division ring with char(D) ̸= 2. Consider
the rings M2(D) and U2(M2(D)). We first note
δ(M2(D)) =M2(δ(D)) =M2(D) and δ(U2(M2(D))) =
[
0 M2(D)
0 M2(D)
]
.
LetA=
[
1 1
0 0
]
,B=
[
1 1
0 0
]
, C=
[
0 0
1 1
]
∈M2(D), X=
[
A B
0 C
]
∈ U2(M2(D)),
A′=
[
1 1
−1 −1
]
∈M2(D), Y=
[
A′ A′
0 0
]
∈ U2(M2(D)), Z=
[
1 0
1 0
]
∈M2(D),
E =
[
Z 0
0 Z
]
∈ Id(U2(M2(D))), T =
[
2 0
−2 0
]
∈M2(D), U =
[
4 0
−4 0
]
∈
M2(D). Set F =
[
T U
0 0
]
∈ U2(M2(D)). Then XY = 0 and Y XE =
F /∈ δ(U2(M2(D))). Note that M2(D) is not reduced since AA′ = 0 and
A′A = A′ ̸= 0.
Theorem 2.12. Let R be a ring with e ∈ Id(R) and R = R/δ(R). If R
is right e-reversible, then R is Zhou right e-reversible.
Proof. Let a, b ∈ R with ab = 0. Then ab = 0 in R. The ring R being
right e-reversible yields bae = 0 entailing that bae ∈ δ(R).
B. Ungor, H. Kose, A. Harmanci 295
Corollary 2.13. Let R be a ring with e ∈ Id(R) and R = R/δ(R). If R
is reversible, then R is Zhou right e-reversible.
Theorem 2.14. Let {Ri}i∈I be a family of rings for a finite index set I,
R =
∏
i∈I
Ri and e
2
i = ei ∈ Ri for each i ∈ I and set e = (ei) ∈ R. Then
Ri is Zhou right ei-reversible for each i ∈ I if and only if R is Zhou right
e-reversible.
Proof. By definition, note that δ(R) =
∏
i∈I
δ(Ri). Assume that Ri is
Zhou right ei-reversible for each i ∈ I. Let a = (ai), b = (bi) ∈ R
with ab = 0. Then aibi = 0. By assumption, biaiei ∈ δ(Ri) for each
i ∈ I. Hence bae ∈ δ(R). Conversely, suppose that R is Zhou right
e-reversible. Let ai, bi ∈ Ri with aibi = 0, where i ∈ I. Consider
a = (. . . , ai, . . . ), b = (. . . , bi, . . . ) ∈ R, where ith entries are ai and bi,
respectively, and other entries are zero. Then ab = 0. By supposition,
bae ∈ δ(R). Componentwise equality implies biaiei ∈ δ(Ri). So Ri is
Zhou right ei-reversible for each i ∈ I.
It is worth considering that the Zhou radical commute with infinite
direct products of rings, but the next example shows that this is not true.
Example 2.15. Let F be a field and R =
∏
F denote an infinite direct
product of F . Clearly, δ(F ) = F . On the other hand, since R is not
semisimple, δ(R) ̸= R. This means that δ(
∏
F ) ̸=
∏
δ(F ).
Recall that a ring R is called semiperfect if R/J(R) is semisimple
and idempotents of R/J(R) can be lifted to R. As a consequence of
Theorem 2.14, we obtain under what conditions semiperfect rings are
Zhou e-reversible.
Theorem 2.16. Let R be a semiperfect ring. If R satisfies one of the
following conditions, then it is Zhou right e-reversible for each e ∈ Id(R).
(1) R is commutative.
(2) R is left morphic left quasi-duo.
Proof. If R is commutative semiperfect, then it is a finite direct pro-
duct of local rings by [13, Theorem 23.11]. If R is left morphic left
quasi-duo semiperfect, then it is a finite direct product of local rings by
[6, Proposition 2.15]. In both cases, R is Zhou right e-reversible by
Examples 2.4(2) and Theorem 2.14.
296 An application of the Zhou radical
Let e, f ∈ Id(R). Then e and f are called isomorphic if Re and Rf are
isomorphic as left R-modules, equivalently, eR and fR are isomorphic as
right R-modules.
Theorem 2.17. Let R be a ring and e, f ∈Id(R). If R is Zhou right
e-reversible and e and f are isomorphic, then R is Zhou right f -reversible.
Proof. Let g : Re→ Rf denote the isomorphism of the left R-modules Re
and Rf . Assume that R is a Zhou right e-reversible ring. Then g being an
isomorphism implies that there exists r ∈ R such that f = g(re). Hence
ef = eg(re) = g(ere). Thus ef−f = g(ere)−g(re) = g(ere−re) ∈ δ(R)
due to ere − re ∈ δ(R) by Proposition 2.6. So ef − f ∈ δ(R). Let a,
b ∈ R with ab = 0. By assumption, bae ∈ δ(R). Since δ(R) is an ideal in
R, baef ∈ δ(R). Therefore baf ∈ δ(R).
We close this section by investigating the Zhou right e-reversibility
of corner rings.
Proposition 2.18. Let R be a ring, e ∈ Id(R) with ReR = R and
f ∈ Id(eRe). If R is Zhou right f-reversible, then eRe is Zhou right
f -reversible.
Proof. Assume that R is Zhou right f -reversible. Let a, b ∈ eRe with
ab = 0. By assumption, baf ∈ δ(R). Since baf = ebafe ∈ eδ(R)e and
eδ(R)e = δ(eRe) by [17, Theorem 3.9], we have baf ∈ δ(eRe).
3. Some ring extensions
In this section, we focus on the Zhou e-reversibility of some ring ex-
tensions such as Morita contexts, Dorroh extensions, skew formal power
series rings, special matrix rings and special subrings of a direct product
of rings.
Proposition 3.1. Let M = (R, V,W, S) be a trivial Morita context with
the Morita context ring T (M) =
[
R V
W S
]
. Let E =
[
e x
y f
]
∈ Id(T (M))
and e ∈ Id(R), f ∈ Id(S). Then T (M) is Zhou right E-reversible if and
only if R is Zhou right e-reversible and S is Zhou right f -reversible.
Proof. By Theorem 1.6, δ(T (M)) =
[
δ(R) V
W δ(S)
]
. Assume that T (M)
is Zhou right E-reversible. Let r, r′ ∈ R and s, s′ ∈ S with rr′ = 0
B. Ungor, H. Kose, A. Harmanci 297
and ss′ = 0. Consider A =
[
r 0
0 s
]
, B =
[
r′ 0
0 s′
]
∈ T (M). Then
AB = 0. By assumption, BAE ∈ δ(T (M)). It implies that r′re ∈ δ(R)
and s′sf ∈ δ(S), and so R is Zhou right e-reversible and S is Zhou right
f -reversible. Conversely, let A =
[
r v
w s
]
, B =
[
r1 v1
w1 s1
]
∈ T (M) with
AB = 0. Then rr1 = 0 and ss1 = 0. Since R is Zhou right e-reversible
and S is Zhou right f -reversible, r1re ∈ δ(R) and s1sf ∈ δ(S), respec-
tively. Hence BAE =
[
r1re ∗
∗ s1sf
]
∈ δ(T (M)). Therefore T (M) is
Zhou right E-reversible.
Now we give two direct consequences of Proposition 3.1. Let R and
S be any rings, M an R-S-bimodule and T the formal triangular matrix
ring
[
R M
0 S
]
. It is known that δ(T ) =
[
δ(R) M
0 δ(S)
]
by Theorem 1.6.
Corollary 3.2. Let T =
[
R M
0 S
]
and e ∈ Id(R), f ∈ Id(S). Then for
E =
[
e 0
0 f
]
∈ Id(T ), the ring T is Zhou right E-reversible if and only if
R is Zhou right e-reversible and S is Zhou right f-reversible.
Corollary 3.3. Let R be a ring, n a positive integer, e ∈ Id(R), and
E ∈ Id(Un(R)) with all diagonal entries equal to e. Then R is Zhou
right e-reversible if and only if Un(R) is Zhou right E-reversible.
Proposition 3.4. Let S be an algebra over a ring R and consider the
Dorroh extension D(R,S) of S by R. Let (e, s) ∈ Id(D(R,S)). Then
D(R,S) is Zhou right (e, s)-reversible if and only if R is Zhou right
e-reversible.
Proof. For the necessity, let a, b ∈ R such that ab = 0. Having (e, s) ∈
Id(D(R,S)) yields e ∈ Id(R). Consider (a, 0), (b, 0) ∈ D(R,S). Then
(a, 0)(b, 0) = 0. Since D(R,S) is Zhou right (e, s)-reversible, we have
(b, 0)(a, 0)(e, s) = (bae, bas) ∈ δ(D(R,S)). By Lemma 1.10(1), bae ∈
δ(R). For the sufficiency, let (a, x), (b, y) ∈ D(R,S) with (a, x)(b, y) = 0.
Then ab = 0. Since R is Zhou right e-reversible, bae ∈ δ(R). By Lem-
ma 1.10(1), we have (b, y)(a, x)(e, t) = (bae, ∗) ∈ δ(D(R,S)).
Let R be a ring, σ : R → R a ring homomorphism and R[[x, σ]]
denote the ring of skew formal power series {
∞∑
i=0
aix
i | ai ∈ R}. Addition
298 An application of the Zhou radical
in R[[x, σ]] is usual one and multiplication is defined by xa = σ(a)x.
The ideal < x > is an essential maximal right ideal of R[[x, σ]], and
δ(R[[x, σ]]) ⊆ δ(R)+ < x > by [3, Proposition 3.15]. In general, for a
ring R, J(R[[x]]) = J(R)+ < x >⊆ δ(R[[x]]) ⊆ δ(R)+ < x >. We now
prove that the inclusion is strict in some cases.
Proposition 3.5. For a semisimple ring R, the following inclusion is
strict
δ(R[[x, σ]]) ⊂ δ(R)+ < x > .
Proof. Since R is semisimple, we have δ(R) = R and J(R) = 0. Also,
R[[x, σ]] is not semisimple. Then J(R[[x, σ]]) =< x >⊆ δ(R[[x, σ]]) ⊊
δ(R)+ < x >= R[[x, σ]].
Proposition 3.6. Let R be an abelian ring, σ : R → R a ring homo-
morphism and e ∈ Id(R). If R[[x, σ]] is Zhou right e-reversible, then R
is Zhou right e-reversible.
Proof. Since R is abelian, all idempotents of R[[x, σ]] are contained in R.
Let a, b ∈ R with ab = 0. Consider f(x) = a, g(x) = b ∈ R[[x, σ]], and
so f(x)g(x) = 0. We have g(x)f(x)e ∈ δ(R[[x, σ]]) since R[[x, σ]] is
Zhou right e-reversible. Having δ(R[[x, σ]]) ⊆ δ(R)+ < x > entails that
bae ∈ δ(R). This completes the proof.
There are rings R and positive integers n ≥ 2 such that Mn(R) need
not be Zhou right E-reversible for some E ∈ Id(Mn(R)) as shown below.
Remark 3.7. Let R be a ring which is not semisimple and n ≥ 2 be an
integer. Consider A = e11 + e12 − e21 − e22, B = e12 − e22 ∈Mn(R) and
E = e11 + e12 ∈ Id(Mn(R)). Then AB = 0, but BAE = −e11 − e12 +
e21 + e22 /∈ δ(Mn(R)) since δ(Mn(R)) = Mn(δ(R)). Therefore Mn(R) is
not Zhou right E-reversible.
On the contrast to the ringMn(R), some subrings ofMn(R) are Zhou
right E-reversible for each integer n ≥ 2.
Proposition 3.8. Let R be a ring with e ∈Id(R) and n ≥ 2 an integer.
Then the following are equivalent.
(1) R is Zhou right e-reversible.
(2) Dn(R) is Zhou right E-reversible, where E = eIn.
(3) Vn(R) is Zhou right E-reversible, where E = eIn.
B. Ungor, H. Kose, A. Harmanci 299
Proof. (1) ⇒ (2) and (3): Note that N(Dn(R)) and N(Vn(R)) are ide-
als of the rings Dn(R) and Vn(R), respectively. If R is Zhou right
e-reversible, then (2) and (3) hold by Examples 2.4(5).
(2) ⇒ (3) Obvious.
(3) ⇒ (1) Assume that (3) holds. Let a, b ∈ R with ab = 0. Write
A = aIn and B = bIn. Then AB = 0, and so BAE ∈ δ(Vn(R)). By
Lemma 1.2, we get bae ∈ δ(R).
Let A be a ring, B a subring of A, and consider the ring
R[A,B] = {(a1, a2, . . . , an, b, b, . . . ) | ai ∈ A, b ∈ B, 1 ⩽ i ⩽ n, n ∈ Z+}.
By Theorem 1.12, it is known that δ(R[A,B]) = R[δ(A), δ(A) ∩ δ(B)].
Let e ∈ Id(B) and E = (e, e, e, . . . ) ∈ Id(R[A,B]). We investigate the
Zhou right e-reversibility of R[A,B] in the following result.
Theorem 3.9. Let A be a ring, B a subring of A, e ∈ Id(B) and E =
(e, e, e, . . . ) ∈ Id(R[A,B]). Then the following are equivalent:
(1) A and B are both Zhou right e-reversible.
(2) R[A,B] is Zhou right E-reversible.
Proof. (1)⇒ (2) Let C = (a1, . . . , an, b, b, . . . ), D = (x1, . . . , xm, y, y, . . . )
in R[A,B] with CD = 0. We consider the two cases: n ≥ m and n < m.
Case I. Assume n ≥ m. Then aixi = 0 for i = 1, 2, . . . ,m, am+iy = 0 for
i = 1, 2, . . . , n−m and by = 0. By (1), xiaie ∈ δ(A) for i = 1, 2, . . . ,m,
yam+ie ∈ δ(A) for i = 1, 2, . . . , n−m and ybe ∈ δ(A) ∩ δ(B). It follows
that DCE ∈ δ(R[A,B]).
Case II. Suppose n < m. Then aixi = 0 for i = 1, 2, . . . , n, bxn+i = 0
for i = 1, 2, . . . ,m−n and by = 0. By (1), xiaie ∈ δ(A) for i = 1, 2, . . . , n,
xn+ib ∈ δ(A) for i = 1, 2, . . . ,m− n and ybe ∈ δ(A) ∩ δ(B).
In both cases, we have DCE ∈ δ(R[A,B]).
(2) ⇒ (1) Let a1, a2 ∈ A and b1, b2 ∈ B with a1a2 = 0 and b1b2 = 0.
Set Y = (a1, b1, b1, . . . ) and Z = (a2, b2, b2, . . . ). Then Y Z = 0. By (2),
ZY E ∈ δ(R[A,B]). Hence a2a1e ∈ δ(A) and b2b1e ∈ δ(A) ∩ δ(B), and
so b2b1e ∈ δ(B). This completes the proof.
We now give an example to illustrate Theorem 3.9.
Example 3.10. Let F be a field, A =M2(F ), B = U2(F ) and consider
R[A,B] = {(a1, a2, a3, . . . , an, b, b, b, . . . ) | n ∈ N, ai ∈ A, b ∈ B}.
300 An application of the Zhou radical
Since δ(A) = A and δ(B) = e12F + e22F , we see by Theorem 1.12 that
δ(R[A,B]) = {(a1, a2, . . . , an, b, b, . . . ) | n ∈ N, ai ∈ A, b ∈ δ(B)} =
R[A, δ(B)]. For e11 ∈ Id(B) and E = (e11, e11, e11, . . . ) ∈ Id(R[A,B]),
in the light of Examples 2.8 and Theorem 3.9, R[A,B] is Zhou right
E-reversible.
4. Some Zhou e-reversible subrings of matrix rings
The rings H3(Z,R): Let R be a ring and consider the ring
H3(Z, R) =
n a1 a2
0 a3 a4
0 0 n
| a1, a2, a3, a4 ∈ R,n ∈ Z
with the usual
matrix addition and multiplication. We have the following.
Lemma 4.1. The following hold for a ring R.
(1) N(H3(Z, R)) =
0 a b
0 c d
0 0 0
∈ H3(Z, R) | c ∈ N(R)
.
(2) δ(H3(Z, R)) =
0 R R
0 δ(R) R
0 0 0
.
Proof. It is routine.
Theorem 4.2. A ring R is Zhou right e-reversible for each e ∈ Id(R) if
and only if H3(Z, R) is Zhou right E-reversible for E = e11 + e22e+ e33.
Proof. Clear.
The rings H(s,t)(R): Let R be a ring and s, t ∈ C(R) be invertible
in R. Let
H(s,t)(R) =
a 0 0
c d e
0 0 f
∈M3(R) | a, c, d, e, f ∈ R, a− d = sc, d− f = te
.
Then H(s,t)(R) is a subring of M3(R).
Lemma 4.3. Let A =
a 0 0
c d f
0 0 g
∈ H(1,1)(R). Then
(1) A ∈ δ(H(1,1)(R)) if and only if a, d, g ∈ δ(R).
B. Ungor, H. Kose, A. Harmanci 301
(2) A ∈ Id(H(1,1)(R)) if and only if a, d, g ∈ Id(R).
Proof. It is routine.
Theorem 4.4. Let R be a ring, e ∈ Id(R) and E = eI3 ∈ Id(H(1,1)(R)).
Then R is Zhou right e-reversible if and only if H(1,1)(R) is Zhou right
E-reversible.
Proof. For the necessity, let A =
a 0 0
c d f
0 0 g
and B =
x 0 0
y z u
0 0 v
∈
H(1,1)(R) with AB = 0. Then ax = 0, dz = 0, gv = 0. Since R
is Zhou right e-reversible, by Lemma 4.3(1), {xae, zde, vge} ⊆ δ(R).
Then BAE =
xae 0 0
∗ zde ∗
0 0 vge
∈ δ(H(1,1)(R)). For the sufficiency, let
a, b ∈ R with ab = 0. We write A = aI3 and B = bI3. Then AB = 0.
Since H(1,1)(R) is Zhou right E-reversible, BAE =
bae 0 0
0 bae 0
0 0 bae
∈
δ(H(1,1)(R)). So we get bae ∈ δ(R) by Lemma 4.3(1).
Generalized matrix rings: Let R be a ring and s be a central
element of R. Then
[
R R
R R
]
becomes a ring denoted by Ks(R) with
addition defined componentwise and multiplication defined in [12] by[
a1 x1
y1 b1
] [
a2 x2
y2 b2
]
=
[
a1a2 + sx1y2 a1x2 + x1b2
y1a2 + b1y2 sy1x2 + b1b2
]
.
In [12], Ks(R) is called a generalized matrix ring over R.
Lemma 4.5. For a ring R, we have the following:
(1) If E =
[
e v
w f
]
∈ Id(K0(R)), then e, f ∈ Id(R).
(2) δ(K0(R)) =
{[
a b
c d
]
| a, d ∈ δ(R), b, c ∈ R
}
.
Proof. (1) Let E2 = E =
[
e v
w f
]
∈ K0(R). Then e2 = e and f2 = f
obviously.
302 An application of the Zhou radical
(2) Let δ(K0(R)) =
[
A V
W B
]
. We claim that A,B ⊆ δ(R). Firstly,
assume that A ̸⊆ δ(R). Then there exists an essential maximal right
ideal M of R such that A ̸⊆ M . Consider the right ideal I =
[
M R
R R
]
of K0(R). The maximality of M in R yields the maximality of I in
K0(R). In order to show that I is essential in K0(R), let 0 ̸= α =[
r v
w s
]
∈ K0(R). If r = 0, then 0 ̸= α ∈ I. If r ̸= 0, then there exists
r1 ∈ R such that 0 ̸= rr1 ∈ M by the essentiality of M in R. Hence
0 ̸= α
[
r1 0
0 0
]
∈ I, and so I is essential in K0(R). Thus δ(K0(R)) ⊆ I,
this entails A ⊆ M . This contradiction shows A ⊆ δ(R). By a similar
discussion, we obtain B ⊆ δ(R). It follows δ(K0(R)) =
[
A V
W B
]
⊆[
δ(R) R
R δ(R)
]
. For the reverse inclusion, consider the subsets X = e12R
and Y = e21R of K0(R). Clearly, X and Y are nilpotent ideals in K0(R).
Consequently, we have X,Y ⊆ δ(K0(R)), and so X+Y ⊆ δ(K0(R)). By
[20, Theorem 1.6(2) and Lemma 1.3(1)], X + Y is δ-small in K0(R). By
the fact that δ(R) is δ-small in R, we have that
[
δ(R) R
R δ(R)
]
/(X + Y )
is δ-small in K0(R)/(X + Y ), and so
[
δ(R) R
R δ(R)
]
is δ-small in K0(R)
by [20, Lemma 1.3(1)]. Thus
[
δ(R) R
R δ(R)
]
⊆ δ(K0(R)). Therefore
δ(K0(R)) =
[
δ(R) R
R δ(R)
]
.
Theorem 4.6. Let R be a ring and e, f ∈ Id(R) and E = ee11+ fe22 ∈
Id(K0(R)). Then R is Zhou right e-reversible and Zhou right f -reversible
if and only if K0(R) is Zhou right E-reversible.
Proof. For the necessity, let A =
[
a b
c d
]
, B =
[
u v
t z
]
∈ K0(R) with
AB = 0. Then au = 0, dz = 0. Consider E =
[
e 0
0 f
]
∈ Id(K0(R)).
Since R is Zhou right e-reversible and Zhou right f -reversible, we have
{uae, zdf} ⊆ δ(R). By Lemma 4.5(2), we have BAE =
[
uae ∗
∗ zdf
]
∈
δ(K0(R)). So K0(R) is Zhou right E-reversible.
B. Ungor, H. Kose, A. Harmanci 303
For the sufficiency, suppose that K0(R) is Zhou right E-reversible. Let
a, b ∈ R with ab = 0. Set A =
[
a 0
0 a
]
, B =
[
b 0
0 b
]
∈ K0(R). Having
ab = 0 yields AB = 0. Since K0(R) is Zhou right E-reversible, BAE =[
bae 0
0 baf
]
∈ δ(K0(R)). By Lemma 4.5(2), bae, baf ∈ δ(R). Hence R is
Zhou right e-reversible and Zhou right f -reversible. This completes the
proof.
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Contact information
B. Ungor Department of Mathematics, Ankara
University, Ankara, Turkey
E-Mail: bungor@science.ankara.edu.tr
H. Kose Department of Computer Science, Ankara
University, Ankara, Turkey
E-Mail: handankose@ankara.edu.tr
A. Harmanci Department of Mathematics, Hacettepe
University, Ankara, Turkey
E-Mail: harmanci@hacettepe.edu.tr
Received by the editors: 08.01.2026
and in final form 24.05.2026.
https://doi.org/10.1142/S0219199717500390
https://doi.org/10.1142/S0219498810003999
https://doi.org/10.1080/00927878708823519
https://doi.org/10.1142/S1005386710000842
https://doi.org/10.1142/S1005386710000842
https://doi.org/10.55730/1300-0098.3246
https://doi.org/10.1080/00927872.2012.748327
https://doi.org/10.1007/s10011-000-0305-9
Burcu Ungor, Handan Kose, and Abdullah Harmanci
|
| id | admjournalluguniveduua-article-2462 |
| institution | Algebra and Discrete Mathematics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-09T01:00:11Z |
| publishDate | 2026 |
| publisher | Lugansk National Taras Shevchenko University |
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| spelling | admjournalluguniveduua-article-24622026-07-08T07:55:33Z An application of the Zhou radical to the \(e\)-reversibility of rings Ungor, Burcu Kose, Handan Harmanci, Abdullah reversible ring, \(e\)-reversible ring, Zhou radical, idempotent element, Morita context 16U40, 16U80, 16N40, 16U99, 16S50 Let \(R\) be a ring and \(e\) be an idempotent element of \(R\). The Zhou radical of \(R\) denoted by \(\delta(R)\) is the intersection of maximal essential right ideals of \(R\). In the literature, \(e\)-reversible rings were studied regarding the question of how idempotent elements affect the reversible property of rings. In this paper, we provide an application of the Zhou radical of a ring to the reversibility depending on idempotents. Accordingly, we study rings \(R\) in which \(ab = 0\) implies \(bae\in\delta(R)\) (or \(eba\in\delta(R),\)) where \(a, b\in R\), called Zhou right (or left) \(e\)-reversible rings. Besides studying the structure of Zhou \(e\)-reversible rings, we investigate relations between Zhou \(e\)-reversible rings and some known rings, such as Zhou \(e\)-reduced rings, central reversible rings, NI-rings, semiperfect rings and matrix rings. In addition to these, we determine the Zhou radical of some certain rings, such as Morita context rings and special subrings of a direct product of rings. 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/2462 10.12958/adm2462 Algebra and Discrete Mathematics; Vol 41, No 2 (2026) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2462/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2462/1368 Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | reversible ring \(e\)-reversible ring Zhou radical idempotent element Morita context 16U40 16U80 16N40 16U99 16S50 Ungor, Burcu Kose, Handan Harmanci, Abdullah An application of the Zhou radical to the \(e\)-reversibility of rings |
| title | An application of the Zhou radical to the \(e\)-reversibility of rings |
| title_full | An application of the Zhou radical to the \(e\)-reversibility of rings |
| title_fullStr | An application of the Zhou radical to the \(e\)-reversibility of rings |
| title_full_unstemmed | An application of the Zhou radical to the \(e\)-reversibility of rings |
| title_short | An application of the Zhou radical to the \(e\)-reversibility of rings |
| title_sort | application of the zhou radical to the \(e\)-reversibility of rings |
| topic | reversible ring \(e\)-reversible ring Zhou radical idempotent element Morita context 16U40 16U80 16N40 16U99 16S50 |
| topic_facet | reversible ring \(e\)-reversible ring Zhou radical idempotent element Morita context 16U40 16U80 16N40 16U99 16S50 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2462 |
| work_keys_str_mv | AT ungorburcu anapplicationofthezhouradicaltotheereversibilityofrings AT kosehandan anapplicationofthezhouradicaltotheereversibilityofrings AT harmanciabdullah anapplicationofthezhouradicaltotheereversibilityofrings AT ungorburcu applicationofthezhouradicaltotheereversibilityofrings AT kosehandan applicationofthezhouradicaltotheereversibilityofrings AT harmanciabdullah applicationofthezhouradicaltotheereversibilityofrings |