A note on locally invo-regular rings

In this paper we mainly exhibit that if R is a locally invo-regular ring of characteristic three with no central nilpotent elements then R need not be isomorphic to the field of order three.

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Introduction

Locally invo-regular rings, invo-regular unital rings and quasi invo-regular rings have been studied recently [1-3]. Recall that a ring R is called an invo-regular ring if rvr r  for each R r  . Here R v  is an element satisfying 1 2  v [1-3]. One may note that as per [1] a ring R is called locally invo-regular if rvr r  or    r v r r     1 1 1 for each R r  .

Thus one may easily find that each invo-regular ring is a locally invo-regular ring. In this paper we provide some results related to locally invo-regular rings. We prove that a locally invo-regular ring R of characteristic three with no central nilpotent elements is not necessarily isomorphic to the field of order three. We produce an example of a ring of order nine and characteristic three which is a locally invo- regular ring with no central nilpotent element however it is not isomorphic to the field of order three.

This example works as a counterexample for the following result. Thus the present work improves the results of existing mathematical literature. In [1, theorem 3] it has been noted that if R is a locally invo-regular ring of characteristic three with no central nilpotent elements then R is isomorphic to the field of order three. In addition we exhibit that a ring without central nilpotent elements need not be a ring without non-trivial idempotent element.

It may be noted that a locally invo-regular ring without non-trivial idempotents is a weakly tripotent ring [1]. In the next section we exhibit that the converse of this result is not true. A 3 [4-5]. weakly tripotent ring is a ring in which for each R a  we have a a  3 or   a a    1 1 In this note R is a unital and associative ring. In the next section we produce desired results.

Main Results Proposition 1: A ring without central nilpotent elements need not be a ring without non-trivial idempotent element. Proof: One can easily note that the field 3 Z of order three does not contain any non-zero nilpotent element and it has been considered as a ring without central nilpotent element in [1]. In addition 3 Z does not contain any non trivial idempotent element. Now we shall prove the Maths 2022; 7(5): 08-09 SK Pandey Department of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Vigyan Nagar, Jodhpur, Rajasthan, India SK Pandey Department of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Vigyan Nagar, Jodhpur, Rajasthan, India

Result

of this proposition by means of the following example. https://www.mathsjournal.com Let            2,2 , 1,2 , 0,2 , 2,1 , 1,1 , 0,1 , 2,0 , 1,0 , 0,0  R three. Thus if R is a locally invo-regular ring of characteristic three with no central nilpotent elements then R need not be isomorphic to the field of order three.

1. Danchev PV. Locally Invo-Regular rings, Azerbaijan . One may verify that R is a ring under component wise addition and multiplication modulo three. One may easily find that R does not have central nilpotent elements as it does not have any non-zero nilpotent elements. Clearly   0,1 and  1,0 are non-trivial idempotent elements Journal of Mathematics. 2021;11(1):28-44
2. Danchev PV. Invo-Regular Unital Rings, Ann. Univ. Mariae Curie-Sklodowska, Sect. A-Math. 2018;72:45-53.
3. Danchev PV. Quasi invo-clean rings, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics. 2021 Jul 8:71-80.
4. Breaz S, Cimpean A. Weakly Tripotent Rings, Bull. of R . Therefore a ring without central nilpotent elements need not be a ring without non-trivial idempotent element. Proposition 2: A weakly tripotent invo-regular ring need not be a ring without non-trivial idempotent elements. Proof: Let us consider the above example. One may easily Korean Math. Soc. 2018;55(4):1179-1187,
5. Pandey SK. Some Counterexamples in Ring Theory, arXiv:2203.02274 [math.RA], 2022 Feb 17. verify that each element a of R satisfies a a  3 or   a a    1 1 3 . Therefore R is a weakly tripotent ring. Further we have      0,0 1,1 0,0 0,0      1,0 1,1 1,0 1,0        2,0 2,2 2,0 2,0       0,1 1,1 0,1 0,1   1,1 1,1 1,1 1,1        2,1 2,1 2,1 2,1        0,2 2,2 0,2 0,2       1,2 1,2 1,2 1,2        2,2 2,2 2,2 2,2  Thus R is an invo-regular ring and hence a locally invo- regular ring. Also,   0,1 and  1,0 are non-trivial idempotent elements of R . Therefore R is a weakly tripotent ring which is a locally invo-regular ring with non-trivial idempotent elements. Proposition 3: A locally invo-regular ring R of characteristic three with no central nilpotent elements is not necessarily isomorphic to the field of order three. Proof; In order to prove this result we consider the same example given in the proof of proposition
6. Let            2,2 , 1,2 , 0,2 , 2,1 , 1,1 , 0,1 , 2,0 , 1,0 , 0,0  R . We have already noted that R is a ring under component wise addition and multiplication modulo three. Also, it does not have central nilpotent elements and the characteristic of R is three. It has been seen that R is an invo-regular ring. And each invo-regular ring is a locally invo-regular ring. Hence R is a locally invo-regular ring. Since R is a ring of order nine as well as it is a ring with zero divisors therefore it cannot be isomorphic to the field of order