Int. J. Anal. Appl. (2023), 21:69

A Note on Skew Generalized Power Serieswise Reversible Property

Eltiyeb Ali1,2,∗

1Department of Mathematics, College of Science and Arts, Najran University, KSA
2Department of Mathematics, Faculty of Education, University of Khartoum, Sudan

∗Corresponding author: eltiyeb76@gmail.com

Abstract. The aim of this paper is to introduce and study (S,ω)-nil-reversible rings wherein we call

a ring R is (S,ω)-nil-reversible if the left and right annihilators of every nilpotent element of R are

equal. The researcher obtains various necessary or sufficient conditions for (S,ω)-nil-reversible rings

are abelian, 2-primal, (S,ω)-nil-semicommutative and (S,ω)-nil-Armendariz. Also, he proved that, if

R is completely (S,ω)-compatible (S,ω)-nil-reversible and J an ideal consisting of nilpotent elements

of bounded index ≤ n in R, then R/J is (S,ω̄)-nil-reversible. Moreover, other standard rings-theoretic
properties are given.

1. Introduction

Throughout this paper, all rings are associated with identity unless otherwise stated. We write

P (R), nil(R), Matn(R), Tn(R, ) Sn(R),R[x], End(R) and Aut(R), respectively for the prime radical,

the set of all nilpotent elements of R, full square matrices, upper square triangular matrices for a

positive integer n with entries in R, the subring consisting of all upper square triangular matrices, the

polynomial ring, the monoid of ring endomorphisms of R and the group of ring automorphisms of R.

The purpose of this article is to examine (S,ω)-nil-reversible rings, where (S,≤) is a strictly ordered
monoid and ω : S → End(R) is a monoid homomorphism. A ring R is considered (S,ω)-nil-reversible
if the left and right annihilators of every nilpotent element in R are equal. The author provides various

necessary or sufficient conditions for (S,ω)-nil-reversible rings to be abelian, 2-primal, (S,ω)-nil-

semicommutative and (S,ω)-nil-Armendariz. I have an example to illustrate, a (S,ω)-nil-reversible

ring may not necessarily be (S,ω)-semicommutative or (S,ω)-reversible. Additionally, it is shown that

Received: May 20, 2023.

2020 Mathematics Subject Classification. 16S99, 16D80, 13C99.

Key words and phrases. Armendariz ring; (S,ω)-reversible; ordered monoid (S,≤); semicommutative ring.

https://doi.org/10.28924/2291-8639-21-2023-69
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-69


2 Int. J. Anal. Appl. (2023), 21:69

if R is completely (S,ω)-compatible and J is an ideal consisting of nilpotent elements with bounded

index ≤ n, then R/J is also (S,ω)-nil-reversible. Furthermore, it is proven that a multiplicatively
closed subset of a ring consisting of central non-zero divisors is (S,ω)-nil-reversible if and only if

the entire ring itself is (S,ω)-nil-reversible. The article also covers other standard properties in ring

theory.

A ring R is said to be reversible if xy = 0, then yx = 0, where x,y ∈ R see Cohn [1]. The
article [2] defines a semicommutative ring R as one where xy = 0 implies xRy = 0 for all x,y ∈ R.
Rings with no nonzero nilpotent elements are called reduced rings and are symmetric, reversible, and

semicommutative according to [3, P. 361] and [3, Proposition 1.3]. However, polynomial rings over

reversible rings need not be reversible as shown in [4, Example 2.1]. In [5], strongly reversible rings

are introduced as reversible rings over which polynomial rings are also reversible. A ring R is strongly

reversible if f (x)h(x) = 0 implies h(x)f (x) = 0 for all polynomials f (x),h(x) ∈ R[x]. Reversible
Armendariz rings satisfy this property, but reduced rings may not be strongly reversible in general. A

ring is called a 2-primal ring if its nilradical coincides with its prime radical, and an NI-ring if its upper

nilradical coincides with its set of nilpotent elements. A ring is an NI-ring if and only if its set of

nilpotent elements forms an ideal, while 2-primal rings are NI-rings.

Armendariz ring defined by the reference [2]. If the products two polynomials f (x)g(x) = 0, then

aibj = 0, for all i, j. In our discussion, we use the following terminology: Given non-empty subsets A

and D of a monoid S, an element u0 ∈ AD = {st : s ∈ A,t ∈ D} is considered a single product
element (abbreviated as s.p. element) in AD if it can be expressed singly in the form u = st. The

following definition will be useful in the next section.

Definition 1.1. The article [6] defines an ordered monoid (S,≤) as an artinian narrow unique product
monoid (or a.n.u.p. monoid) if, for any two artinian and narrow subsets A and D of S, there exists

a unique product element in the set AD that is upper principal. A minimal artinian narrow unique

product monoid (or m.a.n.u.p. monoid) is defined as an ordered monoid (S,≤) where, for any two
artinian and narrow subsets A and D of S, there exist minimum elements a ∈ min(A) and b ∈ min(D)
such that their product ab is an upper principal element of the set AD. A monoid is said to be totally

orderable if it can be ordered with a total order ≤, while a quasitotally ordered monoid is one where
the order ≤ can be refined to a strictly total order �.

To start, we revisit the creation of the generalized power series ring, which was initially presented

in [7]. Let (S,≤) be an ordered set. If every strictly decreasing sequence of elements in S is finite,
then (S,≤) is said to be artinian.

Assume that S is a commutative monoid with the operation denoted additively and the neutral

element denoted by 0. For a ring R, (S,≤) be a strictly ordered monoid, and ω : S → End(R) be a
monoid homomorphism. Denote the image of s under ω as ωs = ω(s) for any s ∈ S.



Int. J. Anal. Appl. (2023), 21:69 3

Let F be the set of all functions f : S → R such that the support supp(f ) = {s ∈ S : f (s) 6= 0}
is both artinian and narrow. For any s ∈ S and f ,g ∈ F, the set Xs(f ,g) consisting of all pairs
(u,v) ∈ supp(f )×supp(g) such that s = uv is finite. Therefore, we can define the product f g : S → R
of f and g as follows: if (u,v) /∈ Xs(f ,g) for all (u,v) ∈ supp(f ) × supp(g), then (f g)(s) = 0,
otherwise, (f g)(s) =

∑
(u,v)∈Xs(f ,g) f (u)ωu(g(v)) is conventionally considered to be 0. Using the

previously defined pointwise addition and multiplication, the set F becomes a ring known as the ring

of skew generalized power series with coefficients in R and exponents in S, denoted by [[RS,≤,ω]] (or

simply R[[S,ω]] if the order ≤ is unambiguous), as described in [8]. A subset P ⊆ R is considered to
be S-invariant if it is ωt-invariant for every t ∈ S, meaning that ωt(P ) ⊆ P . For each element d ∈ R
and each element t ∈ S, we have the elements cd and et in [[RS,≤,ω]] defined by

cd(λ) =



d, λ = 1,

0, λ ∈ S\{1},

et(λ) =




1, λ = t,

0, λ ∈ S\{t}.

The mapping d 7→ cd is a ring embedding of R into the ring [[RS,≤,ω]], while the mapping t 7→ et is
a monoid embedding of S into the multiplicative monoid of that same ring. Moreover, we have the

relationship that etcd = cωt(d)et.

2. (S,ω)-nil-reversible rings

In this section, we introduce the concept of (S,ω)-nil-reversible rings, which is a generalization of

both (S,ω)-reversible rings and generalized power series reversible rings. We then utilize this concept

to investigate the relationships between (S,ω)-nil-reversible rings and certain classes of rings.

Definition 2.1. For a ring R, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid homo-
morphism. R is to be (S,ω)-nil-reversible, if f g ∈ [[nil(R)S,≤,ω]], then gf ∈ [[nil(R)S,≤,ω]], for all
f ,g ∈ [[RS,≤,ω]].

Remark 2.2. By definition, it is clear that, skew generalized power series nil-reversible rings are closed

under subrings.

Definition 2.3. In [6] A ring R is said to be S-compatible (or (S,ω)-compatible) if for every element d

in the strictly ordered monoid S, the corresponding endomorphism ωd of R is compatible. Similarly, a

ring R is said to be S-rigid (or (S,ω)-rigid) if for every element d in S, the corresponding endomorphism

ωd of R is rigid. Here, ω : S → End(R) is a monoid homomorphism that maps elements of the monoid
S to endomorphisms of the ring R.

Lemma 2.4. [6] For a ring R, (S,≤) a strictly ordered monoid and ω : S → End(R) a monoid
homomorphism. A ring R is reduced ⇔ [[RS,≤,ω]] is reduced.



4 Int. J. Anal. Appl. (2023), 21:69

Lemma 2.5. [9] For ω : S →End(R) a monoid homomorphism. Any elements r,t ∈ R and d ∈ S.
The following results are correct:

(1) rt ∈ nil(R) ⇔ rωd(t) ∈ nil(R).
(2) rt ∈ nil(R) ⇔ ωd(r)t ∈ nil(R).

We can provide an example of nil-reversible rings of skew generalized power series that do not fall

under the categories of either skew generalized power series reversible or skew generalized power series

semicommutative. It is important to note that skew generalized power series reversible rings are both

skew generalized power series semicommutative and skew generalized power series nil-reversible by

definition. This leads us to speculate that skew generalized power series nil-reversible rings may also

be skew generalized power series semicommutative. However, the following examples disprove this

possibility. To support this claim, we require the following propositions.

Proposition 2.6. For a ring R, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid ho-
momorphism. Suppose R is an (S,ω)-compatible with nil(R) an ideal, then R is (S,ω)-nil-reversible.

Proof. Let f ,g ∈ [[RS,≤,ω]], satisfying f g is nilpotent. There exists a positive integer ` such that
(f g)` = 0, so (f (r)ωr (g(t)))` = 0 for each r,t ∈ S. Then by compatibility f (r)g(t) ∈ nil(R). Hence
g(t)f (r) is nilpotent. Thus, gf is nilpotent. �

Proposition 2.7. For a ring R, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid
homomorphism. Suppose R is an (S,ω)-compatible. A ring R is (S,ω)-nil-reversible ring if and only

if for any n, the n-by-n upper triangular matrix ring Tn(R) is (S,ω)-nil-reversible.

Proof. Assume that f ,g ∈ [[Tn(R)S,≤,ω]], satisfying f g ∈ [[nil(Tn(R))S,≤,ω]]. So by [10],

nil(Tn(R)) =




nil(R) R R · · · R
0 nil(R) R · · · R
0 0 nil(R) · · · R
...

...
...

...
...

0 0 0 · · · nil(R)



.

If R is a ring with no nonzero nilpotent elements, then the nilradical of R is trivial, i.e., nil(R) = 0.

Therefore, the nilradical of the n-th triangular matrix ring over R, denoted by Tn(R), is also trivial.

Hence, nil(Tn(R)) forms an ideal in Tn(R). By Proposition 2.6, Tn(R) is (S,ω)-nil-reversible. The if

part follows Remark 2.2. �

Example 2.8. For a ring R, (S,ω)-compatible, (S,≤) a strictly ordered monoid and ω : S →End(R)
a monoid homomorphism. Let R be (S,ω)-nil-reversible ring. Then

T =





a11 a12 a13

0 a22 a23

0 0 a33


 | aij ∈ R


 .



Int. J. Anal. Appl. (2023), 21:69 5

is (S,ω)-nil-reversible ring by Proposition 2.7. Note that f g = 0, where f = cE23 + cE13es and

g = cE12 + cE22es, but we have gf 6= 0. So T is not (S,ω)-reversible. In fact, T is not (S,ω)-
semiccomutative by [11, Example 2.5] (with n = 3).

Also, let S be an (S,ω)-nil-reversible ring. Then the ring

Rn =







a a12 a13 · · · a1n
0 a a23 · · · a2n
0 0 a · · · a3n
...

...
...

...
...

0 0 0 · · · a



| a,aij ∈ S; n ≥ 3



.

is not (S,ω)-reversible by [11, Example 2.5]. But Rn is (S,ω)-nil-reversible by Proposition 2.7 since

any subring of (S,ω)-nil-reversible ring is (S,ω)-nil-reversible. It is obvious that R4 is not (S,ω)-

semicommutative and it can be proved similarly that Rn is not (S,ω)-semicommutative for n ≥ 5.

Proposition 2.9. For a ring R, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid ho-
momorphism. Assume that R is (S,ω)-nil-reversible and (S,ω)-compatible. Suppose g1,g2, . . . ,gn ∈
[[RS,≤,ω]] satisfying g1g2 · · ·gn ∈ [[nil(R)S,≤,ω]], then g1(v1)g2(v2)
· · ·gn(vn) ∈ nil(R) for all v1,v2, . . . ,vn ∈ S.

Proof. It is clear by the definition. �

Corollary 2.10. For a ring R, (S,≤) a strictly ordered monoid and w : S → End(R) a monoid
homomorphism and R to be S-compatible. The following conditions are equal:

(1) If g1,g2, . . . ,gn ∈ [[RS,≤,ω]] satisfy g1g2 · · ·gn ∈ [[nil(R)S,≤,ω]], then g1(v1)g2(v2)
· · ·gn(vn) ∈ nil(R), for any v1,v2, . . . ,vn ∈ S.
(2) R is NI ring.

Proposition 2.11. For a ring R, (S,≤) a strictly ordered monoid and w : S → End(R) a monoid
homomorphism. Assume R is an (S,ω)-compatible. If R is (S,ω)-nil-reversible, then nil(R[[S,ω]]) ⊆
nil(R)[[S,ω]].

Proof. Let g ∈ nil(R[[S,ω]]), suppose g` = 0 where ` ∈Z+. Then by Proposition 2.9, g(v) ∈ nil(R)
for each v ∈ S. Thus nil(R[[S,ω]]) ⊆ nil(R)[[S,ω]]. �

Proposition 2.12. Let R be a ring, (S,≤) a strictly ordered monoid and w : S → End(R) a monoid
homomorphism. Suppose that R to be S-compatible. If R is (S,ω)-nil-reversible, then

(1) R is abelian.

() R is 2-primal..

Proof. Let R be a (S,ω)-nil-reversible ring.

(1) Let e be an idempotent element of R. For any g(v) ∈ R,v ∈ S,eg(v) − eg(v)e ∈ nil(R). Note



6 Int. J. Anal. Appl. (2023), 21:69

that (eg(v) − eg(v)e)e = 0. By hypothesis, e(eg(v) − eg(v)e) = 0, so eg(v) = eg(v)e. Again,
g(v)e − eg(v)e ∈ nil(R) and e(g(v)e − eg(v)e) = 0. So by (S,ω)-nil-reversibility of R, we have
(g(v)e −eg(v)e)e = 0, that is, g(v)e = eg(v)e. Hence, eg(v) = g(v)e.
(2) Note that P (R) ⊆ nil(R). Suppose g(v) ∈ nil(R). Then there is a positive integer m ≥ 2 such
that (g(v))m = 0. Thus, R(g(v))m−1g(v) = 0. This implies that g(v)R(g(v))m−1 = 0 as R is

(S,ω)-nil-reversible. This yields (Rg(v))m = 0, so g(v) ∈ P (R). �

According to [9], a ring R is called (S,ω)-nil-Armendariz, if whenever f ,g ∈ R[[S,ω]] satisfying
f g ∈ nil(R[[S,ω]]), then f (r)ωr (g(t)) ∈ nil(R) for all r,t ∈ S.

Proposition 2.13. For a ring R, S-compatible, (S,≤) totally ordered monoid and w : S → End(R)
a homomorphism of monoid. Then, every (S,ω)-nil-reversible rings are (S,ω)-nil-Armendariz.

Proof. Suppose 0 6= f ,g ∈ R[[S,ω]] satisfying f g ∈ nil(R[[S,ω]]). Transfinite induction will be applied
to the set that is strictly and totally ordered (S,≤) showing f (r)g(t) ∈ nil(R) for any r ∈ supp(f )
and t ∈ supp(g). In the ≤′ order, let s and d be the smallest elements in supp(f ) and supp(g),
respectively. If r ∈ supp(f ) and t ∈ supp(g) satisfying r + t = s + d, then s ≤′ r and d ≤′ t.
If s <′ r then s + d <′ r + t = s + d, a contradiction. Thus r = s. Similarly, t = d. Hence

0 = (f g)(s + d) =
∑
(r,t)∈Xs+d(f ,g) f (r)ωr (g(t)) = f (s)ωs(g(d)).

Now let w ∈ S with r + t <′ w,f (r)g(t) = 0. We need to show f (r)ωr (g(t)) ∈ nil(R) for
al r ∈ supp(f ) and t ∈ supp(g) with r + t = w. Writing Xw (f ,g) = {(r,t) | r + t = w as
{(ri,ti ) | i = 1, 2, . . . ,n} such that r1 <′ r2 <′ · · · <′ rn. Since S is cancellative, r1 = r2 and
r1 + t1 = r2 + t2 = w imply t1 = t2. Since ≤′ is a strict order, r1 <′ r2 and r1 + t1 = r2 + t2 = w
imply t2 <′ t1. Thus we have tn <′ · · · <′ t2 <′ t1. Now,

0 = (f g)(w) =
∑

(r,t)∈Xw(f ,g)

f (r)ωr (g(t)) =

n∑
i=1

f (ri )ωri (g(ti )). (2.1)

For each i ≥ 2, r1 + ti <′ ri + ti = w. Therefore, using the induction hypothesis, we can conclude that
f (r1)g(ti ) belongs to the nilradical of R. Since R is a 2-primal ring (as shown in Proposition 2.12),

this implies that f (r1)g(ti ) also belongs to the nilradical of R. Thus, by multiplying equation (2.1) on

the right by f (r1)g(t1), we get:( n∑
i=1

f (ri )g(ti )
)
f (r1)g(t1) = f (r1)g(t1)f (r1)g(t1) = 0.

Then (f (r1)g(t1))2 = 0 and so f (r1)g(t1) ∈ nil(R). Now (2.1) becomes
n∑
i=2

f (ri )g(ti ) = 0. (2.2)

By performing a right-hand side multiplication of (2.2) with f (r2)g(t2), we get f (r2)g(t2) = 0. By

following the same method as described above, we can continue this process and establish proof



Int. J. Anal. Appl. (2023), 21:69 7

f (ri )g(ti ) = 0 for all i = 1, 2, . . . ,n. Thus f (r)g(t) ∈ nil(R) with r + t = w. Hence, utilizing
transfinite induction, it follows that f (r)ωr (g(t)) belongs to the set of nilpotent elements in R for

any r ∈ supp(f ) and t ∈ supp(g). �

Lemma 2.14. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homomorphism
w : S → End(R). Suppose that R is compatible with S. We now examine the conditions for R.
(1) R is (S,ω)-nil-reversible.

(2) If AB is a nilpotent set, then so is BA for each subsets A,B in R.

(3) If KZ is nilpotent, then ZK is nilpotent for right ideals (or left) K,Z in R.

Then (1) ⇒ (2) ⇒ (3).

Proof. The proof is analog with the proof of [11, Lemma 3.5] �

Lemma 2.15. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homomorphism
w : S → End(R). Suppose that R is compatible with S. Then every (S,ω)-nil-reversible rings are
(S,ω)-nil-semicommutative.

Proof. Suppose f ,g ∈ R[[S,ω]] satisfying f g ∈ nil(R[[S,ω]]). Then gf ∈ nil(R[[S,ω]]) and
g(t)ωt(h(w)ωw (f (r))) ∈ nil(R) for any r,t,w ∈ S and h(w) ∈ R, so f (t)h(w)g(r) ∈ nil(R)
by compatibility. Thus, f hg ∈ nil(R[[S,ω]]) by [4, Lemma 1.1]. Therefore, R is an (S,ω)-nil-
semicommutative. �

Proposition 2.16. Consider R is an NI ring and a strictly ordered monoid (S,≤) with a monoid
homomorphism w : S → End(R). Suppose that R is compatible with S. If (S,ω)-nil-reversible with
nil(R) is an ideal of R, then

nil(R)[[S,ω]] = nil(R[[S,ω]]).

Proof. Suppose d ∈ nil(R), by Lemma 2.14, RdR is a nilpotent in R. Since R is compatible with S, for
any λ ∈ S, Rωλ(d)R is a nilpotent ideal of R and so ωλ(d) ∈ nil(R). Thus nil(R) is an invariant with
S and so nil(R)[[S,ω]] is an ideal of R[[S,ω]]. By Proposition 2.11, nil(R[[S,ω]]) ⊆ nil(R)[[S,ω]].
Therefore, it is enough to demonstrate that nil(R)[[S,ω]] ⊆ nil([[RS,≤,ω]]).

Suppose f ∈ nil(R)[[S,ω]] then for any r ∈ S, f (r) ∈ nil(R). By Proposition 2.11, there is a
positive integer ` such that r ∈ S, (Rf (r)R)` = 0. Since R is compatible with S, then for any g,h ∈
R[[S,ω]], (gf h)` = 0. I have know, if g ∈ nil(R)[[S,ω]], then g(t) ∈ nil(R). So g ∈ nil(R[[S,ω]]).
Thus, nil(R)[[S,ω]] ⊆ nil(R[[S,ω]]). Therefore, nil(R)[[S,ω]] = nil(R[[S,ω]]). �

Corollary 2.17. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homomorphism
w : S → End(R). Suppose that R is compatible with S and (S,ω)-reversible. Then g is a nil element
of [[RS,≤,ω]] ⇔ f (r) ∈ nil(R) for all r ∈ S.



8 Int. J. Anal. Appl. (2023), 21:69

Proposition 2.18. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homomor-
phism w : S → End(R). Suppose that R is compatible with S. If a subdirect product of (S,ω)-nil-
reversible rings is finite, then it is also an (S,ω)-nil-reversible ring.

Proof. Suppose we have ideals Jk of R and R/Jk is (S,ω̄)-nil-reversible for k = 1, . . . , l such that⋂l
k=1Jk = 0. Assume that f ,g ∈ R[[S,ω]] satisfying f g ∈ nil(R[[S,ω]]). Then, we have f g ∈

nil(R/Jk[[S,ω]]). Since R/Jk is (S,ω̄)-nil-reversible, we have (f (r)g(t))dr,t,k ∈ Jk for all r,t ∈ S and
k = 1, . . . , l, where dr,t,k is the maximum value of dr,t over all ideals. Thus, (f (r)g(t))dr,t ∈

⋂l
k=1Jk =

0, which implies that f (r)g(t) ∈ nil(R) for all r,t ∈ S. Therefore, we have g(t)f (r) ∈ nil(R) as well,
and so gf ∈ nil(R[[S,ω]]) as desired. �

Proposition 2.19. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homo-
morphism w : S → End(R). Suppose that R is compatible with S and e2 = e ∈ R. If R is
(S,ω)-nil-reversible, then so is eRe.

Proof. Suppose cef ce,cegce ∈ (eRe)[[S,ω]] satisfying (cef ce)(cegce) ∈ nil(eRe)[[S,ω]]. Let e be
an idempotent of R. ce is clearly an idempotent element of (eRe)[[S,ω]], ceg = gce for each g ∈
R[[S,ω]]. Then (cef )(ceg) ∈ nil(eR)[[S,ω]]. Since R is (S,ω)-nil-reversible, the elements f g ∈
nil(R)[[S,ω]], and so gf ∈ nil(R)[[S,ω]]. Then there exists ` ∈ N such that ((cef ce)(cegce))` = 0.
Therefore (cegce)(cef ce) ∈ nil(eRe)[[S,ω]]. �

Corollary 2.20. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homomorphism
w : S → End(R). If and only if R is (S,ω)-nil-reversible, then both eR and (1 − e)R are also
(S,ω)-nil-reversible for a central idempotent e of the ring R.

Proof. Assume that eR and (1 −e)R are (S,ω)-nil-reversible. Since the nil skew generalized power
series reversibility property finite direct products preserve the closure property of the set, R ∼= eR ×
(1 −e)R is (S,ω)-nil-reversible. The converse is true by Proposition 2.19. �

In [12], A homomorphic image of a nil-reversible ring may not be nil-reversible, so as (S,ω)-nil-

reversible by the next example.

Example 2.21. Let (S,≤) a strictly ordered monoid and ω : S → End(R) a monoid homomorphism.
Assume that R = D[[S,≤]], where D is a division ring and I =< xy >, where xy 6= yx. As R
is a domain, R is (S,ω)-nil-reversible. Clearly yx ∈ nil(R/I)[[S,ω]] and x(yx) = xyx = 0. But,
(yx)x = yx2 6= 0. This implies R/I is not (S,ω)-nil-reversible.

Definition 2.22. [9] Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homo-
morphism w : S → End(R). To express the concept of a ring being completely compatible with a set
S, we define it as follows: A ring R is said to be completely S-compatible if every ideal J of R yields

an S-compatible quotient ring R/J. In order to refer to the homomorphism ω, we may alternatively

refer to R as completely compatible with S..



Int. J. Anal. Appl. (2023), 21:69 9

It is evident that any ring that is completely (S,ω)-compatible also qualifies as (S,ω)-compatible.

Another way to express the complete (S,ω)-compatibility of a ring R is by stating that for any subset

J of R and elements r and t in R, the condition rt ∈ J is equivalent to rω(t) ∈ J. This description
will be frequently referenced in our discussions.

Theorem 2.23. Let R be a ring, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid
homomorphism. If R is completely S-compatible (S,ω)-nil-reversible and J an ideal consisting of

nilpotent elements of bounded index ≤ n in R, then R/J is (S,ω)-nil-reversible.

Proof. Suppose f̄ , ḡ ∈ (R/J)[[S,ω]] satisfying f̄ ḡ ∈ nil(R/J)[[S,ω]]. Assuming that the order (S,≤)
can be improved to a strict total order ≤ on S, we will utilize transfinite induction on the strictly totally
ordered set (S,≤) to demonstrate that ḡf̄ ∈ nil(R/J)[[S,ω]]. To begin with, demonstrate through
transfinite induction that g(t)f (s) ∈ nil(R) for every s ∈ supp(f ) and t ∈ supp(g). Given that
supp(f ) and supp(g) are non-empty subsets of S, there exist finite and non-empty sets of minimal

elements in supp(f ) and supp(g), respectively. Let s0 and t0 be the minimum elements in the ≤ order
of these sets. By the same of the proof of [9, Theorem 2.25], we need to show f (s0)ωs0(g(t0)) = 0.

Therefore, by transfinite induction, we can proof that f (s)g(t) = 0. Since f̄ ḡ ∈ nil(R/J)[[S,ω]].
Then, there is a positive integer ` ∈N such that (f̄ ḡ)` = 0̄. So (f (s)g(t))` ∈ J, for any s,t ∈ S. Since
J ⊆ nil(R), (f (s)g(t))` = 0. Hence f (s)g(t) ∈ nil(R) by compatibility, so g(t)f (s) ∈ nil(R), by R
is (S,ω)-nil-reversible, gf ∈ nil(R)[[S,ω]]. Thus ḡf̄ ∈ nil(R/J)[[S,ω]]. Therefore R/J is (S,ω)-nil-
reversible. �

In the next, we utilize the prime radical of a ring to provide descriptions of skew generalized power

series that exhibit nil-reversibility.

Corollary 2.24. Let R be a ring, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid
homomorphism. If R is completely (S,ω)-compatible (S,ω)-nil-reversible, then R/P (R) is (S,ω)-

nil-reversible.

Proof. The proof can be derived from Theorem 2.23 due to the fact that all elements in P (R) are

nilpotent. �

Proposition 2.25. Let R be a ring, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid
homomorphism. Let J be a reduced ideal of a ring R such that R/J is (S,ω)-nil-reversible. Then R

is (S,ω)-nil-reversible.

Proof. Suppose f ,g ∈ [[RS,≤,ω]] satisfying f g ∈ nil(R)[[S,ω]]. Then f̄ ḡ ∈ nil(R/J)[[S,ω]] and
so ḡf̄ ∈ nil(R/J)[[S,ω]] since R/J is (S,ω)-nil-reversible. There is a positive integer ` ∈ N and
(f̄ ḡ)` = 0̄. Therefore (f (s)g(t))` ∈ J for any s,t ∈ S. Since J is reduced, we have f (s)g(t) = 0
yields g(t)f (s) = 0. Thus, gf ∈ nil(R)[[S,ω]]. Therefore, R is (S,ω)-nil-reversible. �



10 Int. J. Anal. Appl. (2023), 21:69

3. Weak annihilator of reversible property of skew generalized power series rings

The concept of weak annihilators and its properties were introduced by Ouyang in [13], with a focus

on subsets X of a ring R put NrR(X) = {a ∈ R|Xa ∈ nil(R)} and NlR(X) = {b ∈ R|bX ∈ nil(R)}. It
can be easily calculated that NrR(X) = NlR(X). The set NrR(X) is called the weak annihilator of X.

If R is a NI-ring, it is evident that NrR(X) forms an ideal of R. Additionally, if R is reduced, then we

have rR(X) = NrR(X) = lR(X) = NlR(X), and more information and findings on weak annihilators

can be found in [14].

Our investigation now focuses on the correlation between weak annihilators in a ring R and those

in a skew generalized power series ring [[RS,≤,ω]]. Let R be a ring and γ = C(f ) be the content of f ,

defined as C(f ) = {f (u)|u ∈ supp(f )}⊆ R. As R ' cR, we can equate the content of f with

cC(f ) = {cf (ui)|ui ∈ supp(f )}⊆ [[R
S,≤,ω]].

Then we have two maps φ : NrAnnR(id(R)) → NrAnn[[RS,≤,ω]](id([[RS,≤,ω]])) and ψ :
NlAnnR(id(R)) → NlAnn[[RS,≤,ω]](id([[RS,≤,ω]])) defined by φ(I) = I[[RS,≤,ω]] and ψ(J) =
[[RS,≤,ω]]J for each I ∈ NrAnnR(id(R)) = {NrR(U)|U is an ideal of R} and J ∈ NlAnnR(id(R)) =
{NlR(U)|U is an ideal of R}, respectively. It is evident that φ is a one-to-one function. The subsequent
theorem demonstrates that φ and ψ are both bijective mappings if and only if R is (S,ω)-nil-reversible.

Theorem 3.1. Let R be a ring, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid
homomorphism. If R is reduced and nil(R) is a nilpotent ideal of R, then the following are equivalent:

(1) R is (S,ω)-nil-reversible ring.

(2) The function φ : NrAnnR(id(R)) → NrAnn[[RS,≤,ω]](id([[RS,≤,ω]])) is bijective, where φ(I) =
I[[RS,≤,ω]] for each I ∈ NrAnnR(id(R)).
(3) The function ψ : NlAnnR(id(R)) → NlAnn[[RS,≤,ω]](id([[RS,≤,ω]])) is bijective, where ψ(J) =
[[RS,≤,ω]]J for every J ∈ NlAnnR(id(R)).

Proof. (1)⇒(2) Suppose Y ⊆ [[RS,≤,ω]] and γ = ∪f∈Y C(f ). By Proposition 2.9 it is enough to
prove Nr[[RS,≤,ω]](f ) = NrRC(f )R[[S,ω]] for every f ∈ Y. We know that, if g ∈ NrR[[S,ω]](f ). Then
f g ∈ nil(R)[[S,ω]]. According to the premise f (di )ωdi (g(tj)) ∈ nil(R) for each di ∈ supp(f ) for all
tj ∈ supp(g). For element di ∈ supp(f ) for every tj ∈ supp(g), 0 = f (di )ωdi (g(tj)) = (cf (di)g)(tj)
and it follows that g ∈ NrR ∪di∈supp(f ) cf (di)R[[S,ω]] = NrRC(f )R[[S,ω]]. So NrR[[S,ω]](f ) ⊆
NrRC(f )R[[S,ω]].

Conversely, suppose g ∈ NrRC(f )R[[S,ω]], so cf (di)g ∈ nil(R)[[S,ω]] for all di ∈ supp(f ). Thus,
(cf (di)g)(tj) = f (di )ωdi (g(tj)) ∈ nil(R) for all di ∈ supp(f ) and tj ∈ supp(g). Therefore,

(f g)(s) =
∑

(di,tj)∈Xs(f ,g)

f (di )ωdi (g(tj)) = 0



Int. J. Anal. Appl. (2023), 21:69 11

it is evident that g ∈ NrR[[S,ω]](f ). Hence NrRC(f )R[[S,ω]] ⊆ NrR[[S,ω]](f ) therefore
NrRC(f )R[[S,ω]] = NrR[[S,ω]](f ). So

NrR[[S,ω]](Y ) = ∩f∈Y NrR[[S,ω]](f ) = ∩f∈Y C(f )R[[S,ω]] = NrR(γ)R[[S,ω]].

(2)⇒(1) Assume the elements f ,g ∈ R[[S,ω]] satisfying f g ∈ nil(R)[[S,ω]]. Then g ∈ NrR[[S,ω]](f )
according to the premise NrR[[S,ω]](f ) = γR[[S,ω]] for any right ideal γ of R. Inversely, 0 = f cg(tj) and

for every di ∈ supp(f ), (f cg(tj))(di ) = f (di )g(tj) ∈ nil(R) for every di ∈ supp(f ) and tj ∈ supp(g).
Thus by reduced ring, g(tj)f (di ) ∈ nil(R), then gf ∈ nil(R)[[S,ω]]. Thus, R is (S,ω)-nil-reversible.
The demonstration of the equivalence between (1)⇔(3) follows a similar approach to that used for
proving the equivalence between (1)⇔(2). �

According to [6], a ring R is defined (S,ω)-Armendariz if for tow polynomial f ,g ∈ R[[S,ω]]
satisfying f g = 0, then f (r)ωr (g(t)) = 0 for all r,t ∈ S. Now we given a strong condition under
which R[[S,ω]] is nil-reversible.

Theorem 3.2. Let R be a ring, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid
homomorphism. If R is (S,ω)-compatible. Assume that R is (S,ω)-Armendariz ring, then R is

(S,ω)-nil-reversible if and only if R[[S,ω]] is nil-reversible.

Proof. Assume that R is (S,ω)-nil-reversible. Let f ,g ∈ R[[S,ω]] be such that f g ∈ nil(R)[[S,ω]].
By Proposition 2.16, nil(R)[[S,ω]] = nil(R[[S,ω]]). So f (ri )g(tj) ∈ nil(R) for every r,t ∈ S,∀ i, j. By
condition that R is (S,ω)-Armendariz, f (ri )ωri (g(tj)) = 0, for all i, j. By compatibility nil-reversibility,

g(tj)f (ri ) ∈ nil(R) for all i, j. So, gf ∈ nil(R)[[S,ω]]. Thus, R[[S,ω]] is nil-reversible. The converse
is clear. �

Theorem 3.3. Consider a ring R and a strictly ordered monoid (S,≤) with a monoid homomorphism
w : S → End(R). Suppose that R is compatible with S. Let ∆ denotes a multiplicatively closed
subset of R consisting of central non-zero divisors. Then R is (S,ω)-nil-reversible if and only if ∆−1R

is (S,ω)-nil-reversible.

Proof. Suppose R is (S,ω)-nil-reversible and pi,dj,u,v ∈ R. Let u−1Cpi,v
−1Cdj ∈ ∆

−1R[[S,ω]]

for all i, j satisfying that u−1Cpiv
−1Cdj ∈ nil(∆

−1R[[S,ω]]). Then (u−1Cpiv
−1Cdj )

` = 0 for some

positive integer `. This implies (CpiCdj )
` = 0, so pidj ∈ nil(R) by using Lemma 2.5 freely. For

any u−1Cpi,v
−1Cdj ∈ ∆

−1R[[S,ω]] having the property that (u−1Cpi )(v
−1Cdj ) = 0, we have

(uv)−1CpiCdj = 0,CpiCdj = 0 for every i, j. By condition that, R is (S,ω)-nil-reversible, djpi ∈ nil(R),
so (v−1u−1)CdjCpi = 0 which further yields (v

−1Cdj )(u
−1Cpi ) ∈ nil(∆

−1R[[S,≤]]). Hence ∆−1R is
(S,ω)-nil-reversible. The converse part is clear. �

A McCoy ring is a generalization of a reversible ring, defined as a ring where the equation f (x)g(x) =

0 implies the existence of a non-zero element d such that f (x)d = 0. Left McCoy rings are defined



12 Int. J. Anal. Appl. (2023), 21:69

similarly. McCoy rings are both left and right McCoy rings. It is known that every reversible ring

is McCoy. However, it cannot be assumed that if a ring R is (S,ω)-nil-reversible, then it is also

(S,ω)-McCoy. An example exists that disproves this assumption.

Example 3.4. Assume that R is a reduced ring, a strictly ordered monoid (S,≤) with a monoid
homomorphism w : S → End(R). Let

Tn(R)) =







a11 a12 a13 · · · a1n
0 a22 a23 · · · a2n
0 0 a33 · · · a3n
...

...
...

...
...

0 0 0 · · · ann



| aij ∈ R



.

Then Tn(R) is not (S,ω)-McCoy by the similar as argument of [16, Example 2.6], but Tn(R) to be

(S,ω)-nil-reversible by Proposition 2.7.

As per Lambek [17], a ring R is considered symmetric if for any x,y,z ∈ R, the condition xyz = 0
implies xzy = 0. It can be easily observed that commutative rings are symmetric and symmetric rings

are reversible rings.

Theorem 3.5. Let R be a ring, (S,ω)-compatible and reversible right Noetherian ring, (S,≤) a strictly
ordered monoid with nil(R) is a nilpotent ideal of R and ω : S →End(R) a monoid homomorphism.
The ring R to be (S,ω)-nil-symmetric if and only if so is [[RS,≤,ω]].

Proof. Assume a ring R is (S,ω)-nil-symmetric such that f ,g,h ∈ [[RS,≤,ω]] satisfying f gh ∈
nil([[RS,≤,ω]]). Hence by Proposition 2.9, f (r)g(d)h(t) ∈ nil(R) for any r,d,t ∈ S. By assump-
tion R is nil-symmetric, then f (r)h(t)g(d) ∈ nil(R). For all s ∈ S. Thus

(f hg)(s) =
∑

(r,t,d)∈Xs(f ,h,g)

f (r)ωr (h(t)ωt(g(d))).

So, the reversibility of R, f hg ∈ nil([[RS,≤,ω]]), it follows that [[RS,≤,ω]] is nil-symmetric. On
the other hand, if [[RS,≤,ω]] is nil-symmetric, then R is (S,ω)-nil-symmetric because subrings of

(S,ω)-nil-symmetric rings is to be (S,ω)-nil-symmetric. �

Conflicts of Interest: The author declares that there are no conflicts of interest regarding the publi-

cation of this paper.

References

[1] P.M. Cohn, Reversible Rings, Bull. London Math. Soc. 31 (1999), 641-648. https://doi.org/10.1112/

s0024609399006116.

[2] M.B. Rege, S. Chhawchharia, Armendariz Rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14-17. https:

//doi.org/10.3792/pjaa.73.14.

https://doi.org/10.1112/s0024609399006116
https://doi.org/10.1112/s0024609399006116
https://doi.org/10.3792/pjaa.73.14
https://doi.org/10.3792/pjaa.73.14


Int. J. Anal. Appl. (2023), 21:69 13

[3] J. Lambek, On the Representation of Modules by Sheaves of Factor Modules, Can. Math. Bull. 14 (1971), 359-368.

https://doi.org/10.4153/cmb-1971-065-1.

[4] N.K. Kim, Y. Lee, Extensions of Reversible Rings, J. Pure Appl. Algebra. 185 (2003), 207-223. https://doi.org/

10.1016/s0022-4049(03)00109-9.

[5] G. Yang, Z.K. Liu, On Strongly Reversible Rings, Taiwan. J. Math. 12 (2008), 129-136. https://www.jstor.org/

stable/43833897.

[6] G. Marks, R. Mazurek, M. Ziembowski, A Unified Approach to Various Generalizations of Armendariz Rings, Bull.

Aust. Math. Soc. 81 (2010), 361-397. https://doi.org/10.1017/s0004972709001178.

[7] P. Ribenboim, Noetherian Rings of Generalized Power Series, J. Pure Appl. Algebra. 79 (1992), 293-312. https:

//doi.org/10.1016/0022-4049(92)90056-l.

[8] R. Mazurek, M. Ziembowski, On Von Neumann Regular Rings of Skew Generalized Power Series, Commun. Algebra.

36 (2008), 1855-1868. https://doi.org/10.1080/00927870801941150.

[9] E. Ali, A. Elshokry, Some Results on a Generalization of Armendariz Rings, Asia Pac. J. Math. 6 (2019), 1.

https://doi.org/10.28924/APJM/6-1.

[10] R. Antoine, Nilpotent Elements and Armendariz Rings, J. Algebra. 319 (2008), 3128-3140. https://doi.org/10.

1016/j.jalgebra.2008.01.019.

[11] E. Ali, A. Elshokry, Extended of Generalized Power Series Reversible Rings, Italian J. Pure Appl. Math. In Press.

[12] S. Subba, T. Subedi, Nil-Reversible Rings, (2021). https://doi.org/10.48550/ARXIV.2102.11512.

[13] L. Ouyang, Extensions of Nilpotent P.P. Rings, Bull. Iran. Math. Soc. 36 (2010), 169-184.

[14] L. Ouyang, Special Weak Properties of Generalized Power Series Rings, J. Korean Math. Soc. 49 (2012), 687-701.

https://doi.org/10.4134/JKMS.2012.49.4.687.

[15] P.P. Nielsen, Semi-Commutativity and the Mccoy Condition, J. Algebra. 298 (2006), 134-141. https://doi.org/

10.1016/j.jalgebra.2005.10.008.

[16] S. Yang, X. Song, Extensions of McCoy Rings Relative to a Monoid, J. Math. Res. Exposition. 28 (2008), 659-665.

https://doi.org/10.3770/j.issn:1000-341X.2008.03.028.

[17] J. Lambek, On the Representation of Modules by Sheaves of Factor Modules, Can. Math. Bull. 14 (1971), 359-368.

https://doi.org/10.4153/cmb-1971-065-1.

https://doi.org/10.4153/cmb-1971-065-1
https://doi.org/10.1016/s0022-4049(03)00109-9
https://doi.org/10.1016/s0022-4049(03)00109-9
https://www.jstor.org/stable/43833897
https://www.jstor.org/stable/43833897
https://doi.org/10.1017/s0004972709001178
https://doi.org/10.1016/0022-4049(92)90056-l
https://doi.org/10.1016/0022-4049(92)90056-l
https://doi.org/10.1080/00927870801941150
https://doi.org/10.28924/APJM/6-1
https://doi.org/10.1016/j.jalgebra.2008.01.019
https://doi.org/10.1016/j.jalgebra.2008.01.019
https://doi.org/10.48550/ARXIV.2102.11512
https://doi.org/10.4134/JKMS.2012.49.4.687
https://doi.org/10.1016/j.jalgebra.2005.10.008
https://doi.org/10.1016/j.jalgebra.2005.10.008
https://doi.org/10.3770/j.issn:1000-341X.2008.03.028
https://doi.org/10.4153/cmb-1971-065-1

	1. Introduction
	2. (S, )-nil-reversible rings
	3. Weak annihilator of reversible property of skew generalized power series rings
	References