Rings in Which Every Element Is a Sum of a Nilpotent and Three 7-Potents

Abstract

In this article, we define and discuss strongly nil-clean rings: every element in a ring is the sum of a nilpotent and three 7-potents that commute with each other. We use the properties of nilpotent and 7-potent to conduct in-depth research and a large number of calculations and obtain a nilpotent formula for the constant . Furthermore, we prove that a ring is a strongly nil-clean ring if and only if , where , , , , , and are strongly nil-clean rings with , , , , , and . The equivalent conditions of strongly nil-clean rings in some cases are discussed.

1. Introduction

In Nicholson’s paper [1], the concept of a clean ring was first mentioned. In [2], Nicholson proposed the concept of strongly clean rings: every element in a ring is the sum of an idempotent and a unit that commute. The classical and interesting generalization is to use a nilpotent instead of a unit to obtain strongly nil-clean elements. Diesl described and characterized strongly nil-clean rings in [3]. Every element of a ring is said to be strongly nil-clean if it is the sum of an idempotent and a nilpotent that commute with one another. The authors in [4, 5] determined the structure of strongly nil-clean rings. The deformation and generalization of idempotents in nil-clean rings can be extended to many new classes of rings. In [6], Chen and Sheibani identified the rings for which every element is the sum of a nilpotent and a tripotent that commute with one another. In [7], Ying et al. determined the rings for which every element is the sum of a nilpotent and two tripotents that commute with one another. In [8], Cui and Xia characterized the rings for which every element is the sum of a nilpotent and three tripotents that commute. Diesl unified the symbols in [9], summarized and extended the previous results, and determined the structure of strongly nil-clean rings and the structure of strongly nil-clean rings. So far, scholars at home and abroad have revealed many interesting results in this field. These results extend the theory of algebraic structures, especially the theory of rings. It provides a new perspective for us to understand the internal structure and properties of rings, which is very important to solve the problem of the classification and properties of rings.

Therefore, this paper will continue this kind of research and use 7-potent to replace 3-potent in the [8] to promote and determine strongly nil-clean rings: every element in a ring is a nilpotent and three 7-potents that commute. At the same time, we prove that a ring is a strongly nil-clean ring if and only if , where , , , , , and are strongly nil-clean rings with , , , , , and . The equivalent conditions of strongly nil-clean rings in some cases are discussed.

2. Structure of Strongly -Nil-Clean Rings

Definition 1. An element in a ring is 3-potent if it satisfies .

Definition 2. An element in a ring is 7-potent if it satisfies .

Definition 3. A ring is a strongly nil-clean ring if every element in is the sum of a nilpotent and three 7-potents that commute with one another.

In order to facilitate writing, the following provisions are given: let and write , where , and , , , commute with one another.

Lemma 4. The following equations are true:

Proof. Sinceso the equation (1) is established. Similarly, the equations (2) and (3) are established. AsTherefore, the equation (4) is established. By the equations (1) and (4), we get.So, the equation (5) is established. By the equations (1)–(5), we obtainThus, the equation (6) is established. By the equation (1), we getSo, the equation (7) is established. By the equation (2), we obtainHence, the equation (8) is established. By the equations (3) and (5), we get.Therefore, . According to this equation, we can easily obtainThe above equations are simplified to giveHence, the equation (9) is established.

Lemma 5. The following formulas are nilpotent:

Proof. AsTherefore, is nilpotent. SinceSo, is nilpotent. SinceHence, is nilpotent.

Lemma 6. is nilpotent, where .

Proof. Substituting (8) and (9) into (19) multiplied by givesis nilpotent. Substituting (7), (8), and (9) into (20) multiplied by givesis nilpotent. Multiplying (25) by in combination with (26) givesis nilpotent. That is, is nilpotent.

Lemma 7. is nilpotent, where , A is the following determinant

Proof. AsSo, . Now, multiplying both sides of this formula by and by Lemma 6, we obtainis nilpotent. SinceSo, is nilpotent. Now, multiplying this formula by and by Lemma 6, we obtainis nilpotent. Substituting (30) into (21) multiplied by givesis nilpotent. Multiplying (32) by in combination with (33) givesis nilpotent. AsHence, multiplying (34) by , , , , , respectively. By Lemma 6, we obtainare nilpotent, where coefficient determinant of exactly . So, is nilpotent.

Lemma 8. is nilpotent.

Proof. Multiplying by (34) and by Lemma 7, we obtain is nilpotent.

Lemma 9. A ring is a strongly nil-clean ring if and only if , where , , , , , and are strongly nil-clean rings with , , , , , and .

Proof. The necessity is obvious. To prove sufficiency below, we first prove that is nilpotent. By Lemma 8, let , we deduce that is nilpotent. Let , we deduce that is nilpotent. So, is nilpotent. Therefore, there exists an integer such that . By the Chinese Remainder Theorem, , where , , , , , , and , , , , , are strongly nil-clean rings.

Lemma 10. Let be a strongly nil-clean ring with . The following is established:(1) is nilpotent for all .(2) is nil and is a subdirect product of rings isomorphic to or .

Proof. (see [10], Theorem 3.5).
As and , where , and , , , commute with one another. If is a 7-potent of , then , so . As there exists polynomials such that . So, is nilpotent.

Lemma 11. Let be a ring with . The following is equivalent:(1) is a strongly nil-clean ring.(2) is nilpotent for all .(3)Every element of is the sum of a nilpotent and a tripotent that commute.(4) is nil and is a subdirect product of ’s.

Proof. (see [11], Proposition 2.8).
The implication is clear.
As and , where , and , , , commute with one another. If is a 7-potent of , then , so . As there exist polynomials such that . So, is nilpotent.

Lemma 12. Let be a strongly nil-clean ring with , then is nilpotent for all .

Proof. As and , where , and , , , commute with one another. As there exist polynomials such that . So, is nilpotent.

Lemma 13. Let be a ring with . The following is equivalent:(1) is a strongly nil-clean ring.(2) is nilpotent for all .(3)Every element of is the sum of a nilpotent and three tripotents that commute.(4) is nil and is a subdirect product of ’s.

Proof. (see [8], Lemma 9).
The implication is clear.
As and , where , and , , , commute with one another. As there exist polynomials such that . So, is nilpotent.

Lemma 14 (see [7], Lemma 3.5). Let . If is nilpotent, then there exists a polynomial such that and is nilpotent.

Lemma 15 (see [11], Lemma 2.6). If and is nilpotent, then there exists a polynomial such that and is nilpotent.

Lemma 16. If is a subdirect product of ’s and , then there exist polynomials such that and are tripotents.

Proof. Let be a subdirect product of with , where . Then is a subring of . We write , let be a disjoint union of such that . Let and setOne can show that there exist polynomials such that for . Indeed,So, for . Let , then are tripotents and . WithThus, we have .

Lemma 17. Let be a ring with . The following is equivalent:(1) is a strongly nil-clean ring.(2) is nilpotent for all .(3) is nil and is a subdirect product of ’s.(4)Every element of is the sum of a nilpotent and six tripotents that commute.

Proof. (see [10], Lemma 3.2).
By Lemma 16 and , then there exist polynomials , , , , , such that , where and they are tripotents. As , so are nilpotent. Since and , so . By Lemma 15, there exist tripotents such that are nilpotent. Then, are nilpotent, where commute with one another and .
The implication is clear.
As and , where , and , , , commute with one another. As there exist polynomials such that . So, is nilpotent.

Lemma 18. If is a subdirect product of ’s and , then there exist polynomials such that and , , , , , , , , are tripotents.

Proof. The proof process is the same as Lemma 16.

Lemma 19. Let be a ring with . The following is equivalent:(1) is a strongly nil-clean ring.(2) is nilpotent for all .(3) is nil and is a subdirect product of ’s.(4)Every element of is the sum of a nilpotent and nine tripotents that commute.

Proof. (see [10], Lemma 3.2).
By Lemma 18 and , then there exist polynomials , , , , , , , , such that , where , , , , , , , , and they are tripotents. As , so are nilpotent. Since and , so . By Lemma 15, there exist tripotents such that are nilpotent. Then, are nilpotent, where commute with one another and .
The implication is clear.
As and , where , and , , , commute with one another. As there exist polynomials such that . So, is nilpotent.
We have completed the scheduled promotion. The following are some other properties of rings.

Theorem 20. The following are equivalent for a ring :(1) is nilpotent for all .(2) is strongly nil-clean for all .

Proof. As and . Thus, . By Lemma 14, there exists an idempotent such that and . So is strongly nil-clean.
Since and is strongly nil-clean, then there exists an idempotent element such that . So . As . Therefore, .

Theorem 21. The following are equivalent for a ring :(1) is nilpotent for all .(2) is strongly nil-clean for all .

Proof. As and . Thus, . By Lemma 14, there exists an idempotent such that and . So is strongly nil-clean.
Since and is strongly nil-clean, then there exists an idempotent element such that . So . As . Therefore, .

Theorem 22. The following are equivalent for a ring :(1) is nilpotent for all .(2) is strongly nil-clean for all .

Proof. As and . Thus . By Lemma 14, there exists an idempotent such that and . So is strongly nil-clean.
Since and is strongly nil-clean, then there exists an idempotent element such that . So . As . Therefore, .

3. Conclusion

In this paper, we mainly study strongly nil-clean rings and obtain the following important conclusions: a ring is a strongly nil-clean ring if and only if , where , , , , , and are strongly nil-clean rings with , , , , , and .

Due to difficulties encountered in the calculations and time constraints, we regret that we have not been able to generalize this ring to more general cases. For example, (1) determine the rings for which every element is the sum of a nilpotent and n 7-potents that commute with one another. (2) Determine the rings for which every element is the sum of a nilpotent and three n-potents that commute with one another.

The above ideas provide new perspectives for the study of related rings and deserve further exploration.

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Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (No. 11871181).