Article
| Title: | $\sqrt {J}U$ rings (English) |
| Author: | Saini, Shiksha |
| Author: | Udar, Dinesh |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1347-1359 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We intend to unviel a new class of rings, called $\sqrt {J}U$ rings, if the units of a ring $R$ equal the sum of 1 and an element from $\sqrt {J(R)}$. Recall that $\sqrt {J(R)}$ is a subset of $R$, not necessarily a subring, which equals $\{ z\in R \colon z^n \in R$ for some $n \geq 1 \}$. Both $UU$ and $JU$ rings are $\sqrt {J}U$ rings credited to the fact that nilpotents and $J(R)$ are subsets of $\sqrt {J(R)}$. The properties exhibited by $\sqrt {J}U$ rings are explored in a thorough manner following which its relations with other rings are observed. For instance, $UNJ$ rings are $\sqrt {J}U$ and no matrix ring, when $n > 1$ is $\sqrt {J}U$. We have focused on extensions of $\sqrt {J}U$ rings like $T(R,M)$, $H_{(p,q)}(R)$, $L_{(p,q)}(R)$, Morita context and group rings. (English) |
| Keyword: | $\sqrt {J}U$ ring |
| Keyword: | $JU$ ring |
| Keyword: | $UU$ ring |
| Keyword: | Jacobson radical |
| Keyword: | nilpotent |
| MSC: | 16N20 |
| MSC: | 16S34 |
| MSC: | 16S50 |
| MSC: | 16U60 |
| MSC: | 16U99 |
| DOI: | 10.21136/CMJ.2025.0117-25 |
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| Date available: | 2025-12-20T07:50:27Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153247 |
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| Reference: | [1] Bergman, G. M.: Modules over coproducts of rings.Trans. Am. Math. Soc. 200 (1974), 1-32. Zbl 0264.16017, MR 0357502, 10.1090/S0002-9947-1974-0357502-5 |
| Reference: | [2] Călugăreanu, G.: UU rings.Carpathian J. Math. 31 (2015), 157-163. Zbl 1349.16059, MR 3408811 |
| Reference: | [3] Chen, H., Gurgun, O., Halicioglu, S., Harmanci, A.: Rings in which nilpotents belong to Jacobson radical.An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 62 (2016), 595-606. Zbl 1389.16032, MR 3680238 |
| Reference: | [4] Connell, I. G.: On the group ring.Can. Math. J. 15 (1963), 650-685. Zbl 0121.03502, MR 0153705, 10.4153/CJM-1963-067-0 |
| Reference: | [5] Danchev, P. V.: Rings with Jacobson units.Toyama Math. J. 38 (2016), 61-74. Zbl 1368.16042, MR 3675274 |
| Reference: | [6] Danchev, P. V., Lam, T.-Y.: Rings with unipotent units.Publ. Math. Debr. 88 (2016), 449-466. Zbl 1374.16089, MR 3491753, 10.5486/PMD.2016.7405 |
| Reference: | [7] Haghany, A.: Hopficity and co-Hopficity for Morita contexts.Commun. Algebra 27 (1999), 477-492. Zbl 0921.16002, MR 1668301, 10.1080/00927879908826443 |
| Reference: | [8] Koşan, M. T.: The p.p. property of trivial extensions.J. Algebra Appl. 14 (2015), Article ID 1550124, 5 pages. Zbl 1326.16023, MR 3339810, 10.1142/S0219498815501248 |
| Reference: | [9] Koşan, M. T., Leroy, A., Matczuk, J.: On $UJ$-rings.Commun. Algebra 46 (2018), 2297-2303. Zbl 1440.16022, MR 3799210, 10.1080/00927872.2017.1388814 |
| Reference: | [10] Koşan, M. T., Quynh, T. C., Žemlička, J.: UNJ-rings.J. Algebra Appl. 19 (2020), Article ID 2050170, 11 pages. Zbl 1461.16022, MR 4136741, 10.1142/S0219498820501704 |
| Reference: | [11] Koşan, M. T., Žemlička, J.: Group rings that are UJ rings.Commun. Algebra 49 (2021), 2370-2377. Zbl 1476.16024, MR 4255013, 10.1080/00927872.2020.1871000 |
| Reference: | [12] Kurtulmaz, Y., Halicioglu, S., Harmanci, A., Chen, H.: Rings in which elements are a sum of a central and a unit element.Bull. Belg. Math. Soc. - Simon Stevin 26 (2019), 619-631. Zbl 1431.15019, MR 4042404, 10.36045/bbms/1576206360 |
| Reference: | [13] Lam, T. Y.: A First Course in Noncommutative Rings.Graduate Texts in Mathematics 131. Springer, New York (1991). Zbl 0728.16001, MR 1125071, 10.1007/978-1-4684-0406-7 |
| Reference: | [14] Ma, G., Wang, Y., Leroy, A.: Rings in which elements are sum of a central element and an element in the Jacobson radical.Czech. Math. J. 74 (2024), 515-533. Zbl 07893396, MR 4764537, 10.21136/CMJ.2024.0433-23 |
| Reference: | [15] Müller, M.: Rings of quotients of generalized matrix rings.Commun. Algebra 15 (1987), 1991-2015. Zbl 0629.16013, MR 0909950, 10.1080/00927878708823519 |
| Reference: | [16] Milies, C. Polcino, Sehgal, S. K.: An Introduction to Group Rings.Algebras and Applications 1. Kluwer Academic, Dordrecht (2002). Zbl 0997.20003, MR 1896125, 10.1007/978-94-010-0405-3 |
| Reference: | [17] Wang, Z., Chen, J.: Pseudo Drazin inverses in associative rings and Banach algebras.Linear Algebra Appl. 437 (2012), 1332-1345. Zbl 1262.47002, MR 2942354, 10.1016/j.laa.2012.04.039 |
| Reference: | [18] Zhou, Y.: Generalizations of $UU$-rings, $UJ$-rings and $UNJ$-rings.J. Algebra Appl. 22 (2023), Article ID 2350102, 11 pages. Zbl 1522.16033, MR 4556318, 10.1142/S0219498823501025 |
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