On the relation between one-sided duoness and commutators
This article studies the structure of rings RR over which the 2×22\times 2 upper triangular matrix rings with the same diagonal are right duo, denoted by D2(R){D}_{2}\left(R). We prove that for any right regular element dd of such a ring RR, dRdR contains the ideal of RR generated by all commutators...
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| Language: | English |
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De Gruyter
2024-12-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2024-0118 |
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| author | Kim Nam Kyun Lee Yang |
| author_facet | Kim Nam Kyun Lee Yang |
| author_sort | Kim Nam Kyun |
| collection | DOAJ |
| description | This article studies the structure of rings RR over which the 2×22\times 2 upper triangular matrix rings with the same diagonal are right duo, denoted by D2(R){D}_{2}\left(R). We prove that for any right regular element dd of such a ring RR, dRdR contains the ideal of RR generated by all commutators. It is also proved that for a domain RR, D2(R){D}_{2}\left(R) is right duo if and only if RR is either commutative or a division ring. Moreover, it is proved that if RR is a ring of characteristic 2 such that D2(R){D}_{2}\left(R) is right duo, then RR has an ascending chain of nil ideals NRi⊆NRi+1{N}_{{R}_{i}}\subseteq {N}_{{R}_{i+1}} (i=0,1,…i=0,1,\ldots ) such that NRi+1⁄NRi{N}_{{R}_{i+1}}/{N}_{{R}_{i}} is contained in the center of R⁄NRiR/{N}_{{R}_{i}}. Furthermore, we give a simpler proof to the famous result that if RR is a simple noncommutative ring then RR coincides with its subring generated by all commutators (by Herstein). Finally, we show that if D2(R){D}_{2}\left(R) is right duo over a ring RR, then Sa⊆aSSa\subseteq aS for any a∈Ra\in R, where SS is any of the following: (i) the prime radical, (ii) the Jacobson radical, (iii) the group of all units in RR, and (iv) the set of one-sided zero-divisors. |
| format | Article |
| id | doaj-art-c543f49ab86641609698a5fa3ad15682 |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj-art-c543f49ab86641609698a5fa3ad156822025-02-02T15:46:02ZengDe GruyterOpen Mathematics2391-54552024-12-01221799110.1515/math-2024-0118On the relation between one-sided duoness and commutatorsKim Nam Kyun0Lee Yang1School of Basic Sciences, Hanbat National University, Daejeon 34158, KoreaDepartment of Mathematics, Yanbian University, Yanji 133002, ChinaThis article studies the structure of rings RR over which the 2×22\times 2 upper triangular matrix rings with the same diagonal are right duo, denoted by D2(R){D}_{2}\left(R). We prove that for any right regular element dd of such a ring RR, dRdR contains the ideal of RR generated by all commutators. It is also proved that for a domain RR, D2(R){D}_{2}\left(R) is right duo if and only if RR is either commutative or a division ring. Moreover, it is proved that if RR is a ring of characteristic 2 such that D2(R){D}_{2}\left(R) is right duo, then RR has an ascending chain of nil ideals NRi⊆NRi+1{N}_{{R}_{i}}\subseteq {N}_{{R}_{i+1}} (i=0,1,…i=0,1,\ldots ) such that NRi+1⁄NRi{N}_{{R}_{i+1}}/{N}_{{R}_{i}} is contained in the center of R⁄NRiR/{N}_{{R}_{i}}. Furthermore, we give a simpler proof to the famous result that if RR is a simple noncommutative ring then RR coincides with its subring generated by all commutators (by Herstein). Finally, we show that if D2(R){D}_{2}\left(R) is right duo over a ring RR, then Sa⊆aSSa\subseteq aS for any a∈Ra\in R, where SS is any of the following: (i) the prime radical, (ii) the Jacobson radical, (iii) the group of all units in RR, and (iv) the set of one-sided zero-divisors.https://doi.org/10.1515/math-2024-0118right duo ringd2(r) is right duocommutatorright regular element16u8016n40 |
| spellingShingle | Kim Nam Kyun Lee Yang On the relation between one-sided duoness and commutators Open Mathematics right duo ring d2(r) is right duo commutator right regular element 16u80 16n40 |
| title | On the relation between one-sided duoness and commutators |
| title_full | On the relation between one-sided duoness and commutators |
| title_fullStr | On the relation between one-sided duoness and commutators |
| title_full_unstemmed | On the relation between one-sided duoness and commutators |
| title_short | On the relation between one-sided duoness and commutators |
| title_sort | on the relation between one sided duoness and commutators |
| topic | right duo ring d2(r) is right duo commutator right regular element 16u80 16n40 |
| url | https://doi.org/10.1515/math-2024-0118 |
| work_keys_str_mv | AT kimnamkyun ontherelationbetweenonesidedduonessandcommutators AT leeyang ontherelationbetweenonesidedduonessandcommutators |