$\sqrt{J}$-clean rings
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917049342623744 |
|---|---|
| author | Udar, Dinesh Saini, Shiksha |
| author_facet | Udar, Dinesh Saini, Shiksha |
| contents | In this paper, we study a new class of rings, called $\sqrt{J}$-clean rings. A ring in which every element can be expressed as the addition of an idempotent and an element from $\sqrt{J(R)}$ is called a $\sqrt{J}$-clean ring. Here, $\sqrt{J(R)}=\{ z\in R : z^n\in J(R) \ \mathrm{for \ some} \ n \geq 1 \}$ where, $J(R)$ is the Jacobson radical. We provide the basic properties of $\sqrt{J}$-clean rings. We also show that the class of semiboolean and nil clean rings is a proper subclass of the class of $\sqrt{J}$-clean rings, which itself is a proper subclass of clean rings. We obtain basic properties of $\sqrt{J}$-clean rings and give a characterization of $\sqrt{J}$-clean rings: a ring $R$ is a $\sqrt{J}$-clean ring iff $R/J(R)$ is a $\sqrt{J}$-clean ring and idempotents lift modulo $J(R)$. We also prove that a ring is a uniquely clean ring if and only if it is a uniquely $\sqrt{J}$-clean ring. Finally, several matrix extensions like $T_n(R)$ and $D_n(R)$ over a $\sqrt{J}$-clean ring are explored. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\sqrt{J}$-clean rings Udar, Dinesh Saini, Shiksha Rings and Algebras 16N20, 16S34, 16S50, 16U60, 16U99 In this paper, we study a new class of rings, called $\sqrt{J}$-clean rings. A ring in which every element can be expressed as the addition of an idempotent and an element from $\sqrt{J(R)}$ is called a $\sqrt{J}$-clean ring. Here, $\sqrt{J(R)}=\{ z\in R : z^n\in J(R) \ \mathrm{for \ some} \ n \geq 1 \}$ where, $J(R)$ is the Jacobson radical. We provide the basic properties of $\sqrt{J}$-clean rings. We also show that the class of semiboolean and nil clean rings is a proper subclass of the class of $\sqrt{J}$-clean rings, which itself is a proper subclass of clean rings. We obtain basic properties of $\sqrt{J}$-clean rings and give a characterization of $\sqrt{J}$-clean rings: a ring $R$ is a $\sqrt{J}$-clean ring iff $R/J(R)$ is a $\sqrt{J}$-clean ring and idempotents lift modulo $J(R)$. We also prove that a ring is a uniquely clean ring if and only if it is a uniquely $\sqrt{J}$-clean ring. Finally, several matrix extensions like $T_n(R)$ and $D_n(R)$ over a $\sqrt{J}$-clean ring are explored. |
| title | $\sqrt{J}$-clean rings |
| topic | Rings and Algebras 16N20, 16S34, 16S50, 16U60, 16U99 |
| url | https://arxiv.org/abs/2510.25341 |