A Note on Commutative Nil-Clean Corners in Unital Rings
We shall prove that if $R$ is a ring with a family of orthogonal idempotents $\{e_i\}_{i=1}^n$ having sum $1$ such that each corner subring $e_iRe_i$ is commutative nil-clean, then $R$ is too nil-clean, by showing that this assertion is actually equivalent to the statement established by Breaz-C\v{a...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2019-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | http://mathizv.isu.ru/en/article/file?id=1304 |
| _version_ | 1818472292760420352 |
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| author | P.V. Danchev |
| author_facet | P.V. Danchev |
| author_sort | P.V. Danchev |
| collection | DOAJ |
| description | We shall prove that if $R$ is a ring with a family of orthogonal idempotents $\{e_i\}_{i=1}^n$ having sum $1$ such that each corner subring $e_iRe_i$ is commutative nil-clean, then $R$ is too nil-clean, by showing that this assertion is actually equivalent to the statement established by Breaz-C\v{a}lug\v{a}reanu-Danchev-Micu in Lin. Algebra \& Appl. (2013) that if $R$ is a commutative nil-clean ring, then the full matrix ring $\mathbbm{M}_n(R)$ is also nil-clean for any size $n$. Likewise, the present proof somewhat supplies our recent result in Bull. Iran. Math. Soc. (2018) concerning strongly nil-clean corner rings as well as it gives a new strategy for further developments of the investigated theme. |
| first_indexed | 2024-04-14T04:05:03Z |
| format | Article |
| id | doaj.art-94e72713fe9e4332955d8fbe33a1c54a |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-04-14T04:05:03Z |
| publishDate | 2019-09-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-94e72713fe9e4332955d8fbe33a1c54a2022-12-22T02:13:23ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852019-09-0129139https://doi.org/10.26516/1997-7670.2019.29.3A Note on Commutative Nil-Clean Corners in Unital RingsP.V. DanchevWe shall prove that if $R$ is a ring with a family of orthogonal idempotents $\{e_i\}_{i=1}^n$ having sum $1$ such that each corner subring $e_iRe_i$ is commutative nil-clean, then $R$ is too nil-clean, by showing that this assertion is actually equivalent to the statement established by Breaz-C\v{a}lug\v{a}reanu-Danchev-Micu in Lin. Algebra \& Appl. (2013) that if $R$ is a commutative nil-clean ring, then the full matrix ring $\mathbbm{M}_n(R)$ is also nil-clean for any size $n$. Likewise, the present proof somewhat supplies our recent result in Bull. Iran. Math. Soc. (2018) concerning strongly nil-clean corner rings as well as it gives a new strategy for further developments of the investigated theme.http://mathizv.isu.ru/en/article/file?id=1304nil-clean ringsnilpotentsidempotentscorners |
| spellingShingle | P.V. Danchev A Note on Commutative Nil-Clean Corners in Unital Rings Известия Иркутского государственного университета: Серия "Математика" nil-clean rings nilpotents idempotents corners |
| title | A Note on Commutative Nil-Clean Corners in Unital Rings |
| title_full | A Note on Commutative Nil-Clean Corners in Unital Rings |
| title_fullStr | A Note on Commutative Nil-Clean Corners in Unital Rings |
| title_full_unstemmed | A Note on Commutative Nil-Clean Corners in Unital Rings |
| title_short | A Note on Commutative Nil-Clean Corners in Unital Rings |
| title_sort | note on commutative nil clean corners in unital rings |
| topic | nil-clean rings nilpotents idempotents corners |
| url | http://mathizv.isu.ru/en/article/file?id=1304 |
| work_keys_str_mv | AT pvdanchev anoteoncommutativenilcleancornersinunitalrings AT pvdanchev noteoncommutativenilcleancornersinunitalrings |