On Weakly Tripotent and Locally Invo-Regular Rings

Preprint

Article

On Weakly Tripotent and Locally Invo-Regular Rings

Altmetrics

Downloads

138

Views

Comments

S K Pandey^*

This version is not peer-reviewed

Submitted:

24 October 2023

Posted:

25 October 2023

You are already at the latest version

Alerts

Abstract

In this article some important observations have been reported on recent works related to weakly tripotent rings and locally invo-regular rings. Our findings give additional results as well as correct some recent results on weakly tripotent rings and locally invo-regular rings appeared in Rendiconti Sem. Mat. Univ. Pol. Torino (2021) and Azerbaijan Journal of Mathematics (2021) respectively. In addition we exhibit that if the Jacobson radical J(A) of a ring is strongly involution t-clean then the characteristic of J(A) need not be four. This finding improves an important result appeared in Eur. J. Pure Appl. Math (2022).

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

In this paper

A

is a unital and associative ring and

J (A)

and

U (A)

stand for the Jacobson radical of

A

and the set of units in

A

respectively. We denote the set of all nilpotents and the set of all idempotents in

A

N (A)

and

I d (A)

respectively. We recall that a ring

A

is said to be a weakly tripotent ring if

u^{3} = u

{(1 - u)}^{3} = 1 - u

for each

u \in A

[1,2] and a ring

A

is said to be a locally invo-regular ring if

u = u v u

1 - u = (1 - u) v (1 - u)

for each

u \in A

and some

v \in A

with

v^{2} = 1

[3].

A ring

A

is called an involution clean ring if every element of

A

is expressible as

a + b

for some

a \in I n v (A)

and some

b \in I d (A)

. If

a b = b a

, then

A

is called strongly involution clean ring [4,5]. A ring

A

is called an involution t-clean ring if every element of

A

is expressible as

u + t

for some

u \in I n v (A)

and some

t \in T r i p (A)

. If

u t = t u

, then

A

is called a strongly involution t-clean ring [4]. It directly follows from these definitions that each involution-clean ring is an involution t-clean ring.

It may be worth mentioning that weakly tripotent rings, locally ino-regular rings and associated notions have extensively appeared in mathematical literature [1,2,3,4,5,6,7,8,9,10]. Motivated by some of our recent works [11,12], here we take an opportunity to report some significant observations and results on weakly tripotent and locally invo-regular rings. In addition we provide some significant results on involution t-clean rings.

In [2] it has been seen that if

A

is a weakly tripotent ring having no non-trivial idempotents and

2

is nilpotent in

A

then

\frac{A}{J (A)} ≅ Z_{2}

and

u^{2} = 2 u = 0

holds for each

u \in J (A)

. Similarly it has been seen in [3] that if

A

is a locally invo-regular ring having no non-trivial idempotents and

2

is nilpotent in

A

then

\frac{A}{J (A)} ≅ Z_{2}

and

u^{2} = 2 u = 0

holds for each

u \in J (A)

However we observe that if

A

is a weakly tripotent ring and it does not have non-trivial idempotents and

2

is nilpotent in

A

then

u^{2} = 2 u = 0

is not necessarily true for each

u \in J (A)

. Similarly we note that if

A

is a locally invo-regular ring having no non-trivial idempotents and

2

is nilpotent in

A

then

u^{2} = 2 u = 0

is not necessarily true for each

u \in J (A)

Moreover we observe that if

A

is a weakly tripotent (or locally invo-regular) ring having no non-trivial idempotents such that

u^{2} = 2 u = 0

for each

u \in J (A)

then

u^{3} = 4 u = 0

for each

u \in J (A)

but the converse of this result is not valid. We exhibit that if

A

is a weakly tripotent (or locally invo-regular) ring having no non-trivial idempotents and

2

is nilpotent in

A

, then

u^{3} = 4 u = 0

for each

u \in J (A)

Further as per [4, Proposition 2.10], if

R

is a ring such that

J (R)

is strongly involution t-clean, then the characteristic of

J (R)

is four. However we exhibit that if

R

is a ring such that

J (R)

is strongly involution t-clean then the characteristic of

J (R)

need not be four.

We provide our observations and results in the next section.

2. Some Observations and Results

Theorem 2.1.

Let

A

is a weakly tripotent ring having no non-trivial idempotents and

2

is nilpotent in

A

, then

u^{3} = 4 u = 0

for each

u \in J (A)

Proof.

Let

A

is a weakly tripotent ring having no non-trivial idempotents and

2

is nilpotent in

A

. By [1, Corollary 10], we have

u^{2} = 1

for each

u \in U (A)

and

u^{2} = 2 u

for each

u \in J (A)

. It may be noted that if

u \in J (A)

then

1 + u \in U (A)

. Similarly

1 - u \in U (A)

. We note that

1 + u \in U (A)

gives that

{(1 + u)}^{2} = 1 \Rightarrow u^{2} = - 2 u

and

1 - u \in U (A)

gives that

{(1 - u)}^{2} = 1 \Rightarrow u^{2} = 2 u

. Hence

u^{2} = - 2 u

and

u^{2} = 2 u

together give that

u^{3} = 4 u = 0

for each

u \in J (R A)

Theorem 2.2.

Let

A

is a locally invo-regular ring having no non-trivial idempotents and

2

is nilpotent in

A

, then

u^{3} = 4 u = 0

for each

u \in J (A)

Proof.

The proof of this Theorem follows from the proof of Proposition 2.1 and the fact that each weakly tripotent ring is a locally invo-regular ring [ 3].

Proposition 2.3.

Let

A

is a weakly tripotent ring having no non-trivial idempotents and

2

is nilpotent in

A

then

u^{2} = 2 u = 0

is not necessarily true for each

u \in J (A)

Proof.

Let

A = Z_{4}

and

G = \{1, g : g^{2} = 1\}

. Clearly

G

is an abelian group under multiplication. Now we shall construct the group ring

A G

. It may be noted that if

a_{i} \in A, g_{i} \in G

then

u \in A G

is expressible as

(a_{1} g_{1} + a_{2} g_{2} + ... + a_{n} g_{n}) \in A G

[13]. Thus the group ring

A G

has the following sixteen elements.

0, 1, 2, 3, g, 2 g, 3 g, 1 + g, 2 + g, 3 + g, 1 + 2 g, 2 + 2 g, 3 + 2 g, 1 + 3 g, 2 + 3 g, 3 + 3 g .

One may easily note that each element

u \in A G

satisfies

u^{3} = u

{(1 - u)}^{3} = 1 - u

. Hence

A G

is a weakly tripotent ring. We note that

0

and

1

are idempotent elements of

R

and

R

does not have any other idempotent element. Also

2

is nilpotent in

R

. We have

U (A) = \{1, 3, g, 2 + g, 1 + 2 g, 3 + 2 g, 2 + 3 g\}

and

J (A) = \{0, 2, 2 g, 3 + g, 2 + 2 g, 1 + 3 g, 3 + 3 g\}

Clearly

3 + 3 g \in J (A)

, but

{(3 + 3 g)}^{2} = 2 (3 + 3 g) \neq 0

. Hence the proof is complete.

Proposition 2.4.

Let

A

is a locally invo-regular ring having no non-trivial idempotents and

2

is nilpotent in

A

then

u^{2} = 2 u = 0

is not necessarily true for each

u \in J (A)

Proof.

We prove it as follows. Let us consider the ring

A

given above (we refer the proof of Proposition 2.3). After some computation one finds that

u = u v u

1 - u = (1 - u) v (1 - v)

holds for each

u \in A

and some

v \in A

with

v^{2} = 1

. Therefore

A

is a locally invo-regular ring.

We have already noted that

2

is nilpotent in

A

and

A

is has no non-trivial idempotent elements. Further

1 + u \in J (A)

such that

{(1 + u)}^{2} = 2 (1 + u) \neq 0

. Hence the proof is complete.

Proposition 2.5.

Let

A

is a weakly tripotent ring having no non-trivial idempotents then

u^{2} = 2 u = 0 \Rightarrow

u^{3} = 4 u = 0

for each

u \in J (A)

but the converse of this result is not valid.

Proof.

Let

A

is a weakly tripotent ring such that it has no non-trivial idempotents. Let

u^{2} = 2 u = 0

for each

u \in J (A)

. This gives that

u^{3} = 2 u^{2} = 0

. This in turn implies that

u^{3} = 4 u = 0

for each

u \in J (A)

The converse is not valid. Let us consider the ring

A

given in the proof of Proposition 2.3. Clearly

1 + u \in J (R)

such that

{(1 + u)}^{3} = 4 (1 + u) = 0

but

{(1 + u)}^{2} = 2 (1 + u) \neq 0

Proposition 2.6.

Let

A

is a locally invo-regular ring having no non-trivial idempotents then

u^{2} = 2 u = 0 \Rightarrow

u^{3} = 4 u = 0

for each

u \in J (A)

but the converse of this result is not valid.

Proof.

The proof directly follows from the above.

Proposition 2.7.

Let

A

is a noncommutative ring such that

J (A)

is strongly involution t-clean, then the characteristic of

J (A)

is not necessarily four

Proof.

Let

A = \{a_{0}, e, a, b, e + a, e + b, a + b, e + a + b\}

. Here we take

a_{0} = (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}), e = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), a = (\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}), b = (\begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix}), e + b = (\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}), e + a = (\begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix}), e + a + b = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), a + b = (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}) .

One may verify that

A

is a noncommutative ring with identity under addition and multiplication of matrices modulo two. We have

N (A) = J (A) = \{a_{0}, a + b\} .

I d (A) = \{a_{0}, e, e + a, e + b, a, b\} .

I n v (A) = \{e, e + a + b\} = U (A) .

T r i p (A) = \{e, e + a + b\} \cup I d \{A\} .

It is clear that

a_{0}

is a strongly involution t- clean element. It may be noted that

a + b = u + t

such that

u \in I n v (A)

t \in T r i p (A)

and

u t = t u

. Here

u = e

and

t = (e + a + b)

. Therefore

a + b

is also strongly involution t-clean element. Thus

J (A)

is strongly involution t-clean. However the characteristic of

J (A)

is not four. Hence our claim is verified.

Proposition 2.8.

Let

A

is a commutative ring such that

J (A)

is strongly involution t-clean, then the characteristic of

J (A)

is not necessarily four

Proof.

Let

A = \{a_{0}, e, a, e + a\}

. Here we take

a_{0} = (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}), e = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), a = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), e + a = (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}) .

It is easy to check that

A

is a commutative ring with identity under the addition and the multiplication of matrices modulo two. Clearly in this case we have

I n v (A) = \{e, a\} = U (A) .

I d (A) = \{a_{0}, e\}

T r i p (A) = I n v (A) \cup \{a_{0}\} and N (A) = J (A) = \{a_{0}, e + a\} .

One may verify that

J (A)

is strongly involution t-clean. However the characteristic of

J (A)

is not four. Hence our claim is justified.

Proposition 2.9.

Let

A

is a ring such that

J (A)

is strongly involution t-clean, then

a^{2} = 2 a = 0

for each

a \in J (A)

implies

a^{3} = 4 a = 0

for each

a \in J (A)

. The converse is not true.

Proof.

The proof easily follows. The converse is not valid can be seen as follows. Let

A = Z_{8}

. It is easy to see that

A

is an involution t-clean ring. It may be noted that

a^{3} = 4 a = 0

for each

a \in J (A)

. However

a^{2} = 2 a = 0

does not hold for each

a \in J (A)

In the following Proposition we prove that if

A

is any ring such that

J (A)

is strongly involution t-clean, then the characteristic of

J (A)

is two provided

a^{2} = 0

for each

a \in J (A)

Proposition 2.10.

A

is a ring such that

J (A)

is strongly involution t-clean, then the characteristic of

J (A)

is two provided

a^{2} = 0

for each

a \in J (A)

Proof.

Let

A

is a ring and

J (A)

is strongly involution t-clean. Let

a^{2} = 0

for each

a \in J (A)

. Then by Proposition 7 [4], there exist

u \in I n v (A)

and

t \in I n v (A)

such that

a = u + t

. This gives

2 + 2 u t = 0 \Rightarrow 2 t (t + u) = 2 a t = 0

. Finally

2 a t = 0

gives

2 a = 0

(since

t^{2} = 1

). Therefore

2 a = 0

for each

a \in J (A)

. Hence the characteristic of

J (A)

is two in this case. Thus proof is complete.

Conflict of Interest

The author declares that there is no conflict of interest.

References

Breaz, S., Cimpean, A., Weakly tripotent rings. Bull. Korean Math. Soc. 2018, 55, 1179–1187.
Danchev, P., A Characterization of Weakly Tripotent Rings, Rendiconti Sem. Mat. Univ. Pol. Torino 2021, 79, 21–32.
Danchev, P. V. Locally Invo-Regular Rings. Azerbaijan Journal of Mathematics 2021, 11. [Google Scholar]
Al.Neima, Mohammed, et al., Involution t-clean rings with applications. Eur. J. Pure Appl. Math 2022, 15, 1637–1648. [CrossRef]
Danchev, P. V., Invo-clean unital rings. Commun. Korean Math. Soc 2017, 32, 19–27. [CrossRef]
Zhou, Y. Rings in which elements are sums of nilpotents, idempotents and tripotents. J. Algebra Appl. 2018, 17. [Google Scholar] [CrossRef]
P. V. Danchev, Quasi invo-clean rings, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics 2021, 63.
Ying, Zhiling, Kosan, Tamer, Zhou, Yiqiang Rings in which every element is a sum of two tripotents. Canad. Math. Bull. 2016, 59, 661–672. [CrossRef]
G. Calugareanu, Tripotents: A Class of Strongly, Clean Elements in Rings. An. St. Univ., Ovidius Constanta 2018, 26, 69–80.
P. Danchev, Commutative Weakly Tripotent Group Rings. Bul. Acad. Stiinte Repub. Mold. Mat. 2020, 93, 24–29.
Pandey, S. K., Some counterexamples in ring theory, arXiv:2203.02274 [math.RA], 2022.
Pandey, S. K., A note on rings in which each element is a sum of two idempotents, Elem. Math. (2023). [CrossRef]
Passman, D. S. The Algebraic Structure of Group Rings, Dover Publications, 2011.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.