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On Weakly Tripotent and Locally Invo-Regular Rings

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Submitted:

14 October 2023

Posted:

16 October 2023

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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.
Keywords: 
tripotent ring; weakly tripotent ring; locally invo-regular rings
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 recall that a ring A is said to be a weakly tripotent ring if u 3 = u or ( 1 − u ) 3 = 1 − u for each u ∈ A [1-2] and a ring A is said to be a locally invo-regular ring if u = u v u or 1 − u = ( 1 − u ) v ( 1 − u ) for each u ∈ A and some v ∈ A with v 2 = 1 [3].
It may be worth mentioning that weakly tripotent rings, locally ino-regular rings and associated notions have extensively appeared in mathematical literature [1-10]. 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 [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 A J ( A ) ≅ Z 2 and u 2 = 2 u = 0 holds for each u ∈ 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 A J ( A ) ≅ Z 2 and u 2 = 2 u = 0 holds for each u ∈ 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 ∈ 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 ∈ 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 ∈ J ( A ) then u 3 = 4 u = 0 for each u ∈ 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 ∈ J ( A ) .
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 ∈ 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 ∈ U ( A ) and u 2 = 2 u for each u ∈ J ( A ) . It may be noted that if u ∈ J ( A ) then 1 + u ∈ U ( A ) . Similarly 1 − u ∈ U ( A ) . We note that 1 + r ∈ U ( R ) gives that ( 1 + r ) 2 = 1 ⇒ r 2 = − 2 r and 1 − u ∈ U ( A ) gives that ( 1 − u ) 2 = 1 ⇒ 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 ∈ 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 ∈ 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 ∈ 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 ∈ A , g i ∈ G then u ∈ A G is expressible as ( a 1 g 1 + a 2 g 2 + ... + a n g n ) ∈ 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 ∈ A G satisfies u 3 = u or ( 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 ∈ J ( A ) , but ( 3 + 3 g ) 2 = 2   ( 3 + 3 g ) ≠ 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 ∈ 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 or 1 − u = ( 1 − u ) v ( 1 − v ) holds for each u ∈ A and some v ∈ 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 ∈ J ( A ) such that ( 1 + u ) 2 = 2   ( 1 + u ) ≠ 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 ⇒ u 3 = 4 u = 0 for each u ∈ 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 ∈ J ( A ) . This gives that u 3 = 2 u 2 = 0 . This in turn implies that u 3 = 4 u = 0 for each u ∈ J ( A ) The converse is not valid. Let us consider the ring A given in the proof of Proposition 2.3. Clearly 1 + u ∈ J ( R ) such that ( 1 + u ) 3 = 4 ( 1 + u ) = 0 but ( 1 + u ) 2 = 2 ( 1 + u ) ≠ 0 .
Proposition 2.6: 
Let A is a locally invo-regular ring having no non-trivial idempotents then u 2 = 2 u = 0 ⇒ u 3 = 4 u = 0 for each u ∈ J ( A ) but the converse of this result is not valid.
Proof. 
The proof directly follows from the above.

Conflicts of Interest

The author declares that there is no conflict of interest.

References

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