Article
| Title: | Unit nil-clean and singular clean group rings (English) |
| Author: | Sharma, Rajendra Kumar |
| Author: | Singh, Amit B. |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 645-653 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We study the unit nil-cleanness of group rings when $R$ is commutative or arbitrary. Furthermore, we investigate some properties of singular clean group rings. A necessary and sufficient condition for the group ring $RG$ to be singular clean is provided. (English) |
| Keyword: | unit nil-clean ring |
| Keyword: | clean ring |
| Keyword: | $p$-group |
| Keyword: | locally finite group |
| Keyword: | singular clean ring |
| Keyword: | UU ring |
| Keyword: | nil-clean ring |
| MSC: | 16S34 |
| MSC: | 16U99 |
| DOI: | 10.21136/CMJ.2025.0371-24 |
| . | |
| Date available: | 2025-05-20T11:49:55Z |
| Last updated: | 2025-05-26 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152963 |
| . | |
| Reference: | [1] Abdolyousefi, M. S., Ashrafi, N., Chen, H.: On unit nil-clean rings.Mediterr. J. Math. 16 (2019), Article ID 100, 9 pages. Zbl 1412.16029, MR 3973269, 10.1007/s00009-019-1369-z |
| Reference: | [2] Amini, A., Amini, B., Nejadzadeh, A., Sharif, H.: Singular clean rings.J. Korean Math. Soc. 55 (2018), 1143-1156. Zbl 1396.16032, MR 3849355, 10.4134/JKMS.j170615 |
| Reference: | [3] Călugăreanu, G.: UU rings.Carpathian J. Math. 31 (2015), 157-163. Zbl 1349.16059, MR 3408811 |
| Reference: | [4] Chen, J., Li, Y., Zhou, Y.: Morphic group rings.J. Pure Appl. Algebra 205 (2006), 621-639. Zbl 1094.16018, MR 2210221, 10.1016/j.jpaa.2005.07.021 |
| Reference: | [5] Chen, J., Nicholson, W. K., Zhou, Y.: Group rings in which every element is uniquely the sum of a unit and an idempotent.J. Algebra 306 (2006), 453-460. Zbl 1110.16025, MR 2271346, 10.1016/j.jalgebra.2006.08.012 |
| Reference: | [6] Chen, J., Zhou, Y.: Strongly clean power series rings.Proc. Edinb. Math. Soc., II. Ser. 50 (2007), 73-85. Zbl 1128.16030, MR 2294005, 10.1017/S0013091505000404 |
| Reference: | [7] Connell, I. G.: On the group ring.Can. J. Math. 15 (1963), 650-685. Zbl 0121.03502, MR 0153705, 10.4153/CJM-1963-067-0 |
| Reference: | [8] Davis, K. S., Webb, W. A.: Lucas' theorem for prime powers.Eur. J. Comb. 11 (1990), 229-233. Zbl 0704.11002, MR 1059553, 10.1016/S0195-6698(13)80122-9 |
| Reference: | [9] Dickson, L. E.: History of the Theory of Numbers. Vol. 1.Chelsea, New York (1952). MR 0245500 |
| Reference: | [10] Diesl, A. J.: Nil clean rings.J. Algebra 383 (2013), 197-211. Zbl 1296.16016, MR 3037975, 10.1016/j.jalgebra.2013.02.020 |
| Reference: | [11] Fine, N. J.: Binomial coefficients modulo a prime.Am. Math. Mon. 54 (1947), 589-592. Zbl 0030.11102, MR 0023257, 10.2307/2304500 |
| Reference: | [12] Guo, Y., Guo, X.: Semigroup rings which are strongly nil-clean.(to appear) in J. Algebra Appl. 10.1142/S0219498825502457 |
| Reference: | [13] Ilić-Georgijević, E., Şahinkaya, S.: On graded nil clean rings.Commun. Algebra 46 (2018), 4079-4089. Zbl 1430.16042, MR 3820617, 10.1080/00927872.2018.1435791 |
| Reference: | [14] Karpilovsky, G.: The Jacobson radical of commutative group rings.Arch. Math. 39 (1982), 428-430. Zbl 0487.16012, MR 0688694, 10.1007/BF01899543 |
| Reference: | [15] Kelarev, A. V.: Ring Constructions and Applications.Series in Algebra 9. World Scientific, Singapore (2002). Zbl 0999.16036, MR 1875643, 10.1142/4807 |
| Reference: | [16] Lam, T. Y.: A First Course in Noncommutative Rings.Graduate Texts in Mathematics 131. Springer, New York (2001). Zbl 0980.16001, MR 1838439, 10.1007/978-1-4419-8616-0 |
| Reference: | [17] McGovern, W. Wm., Raja, S., Sharp, A.: Commutative nil clean group rings.J. Algebra Appl. 14 (2015), Article ID 1550094, 5 pages. Zbl 1325.16024, MR 3338090, 10.1142/S0219498815500942 |
| Reference: | [18] Nicholson, W. K.: Lifting idempotents and exchange rings.Tran. Am. Math. Soc. 229 (1977), 269-278. Zbl 0352.16006, MR 0439876, 10.1090/S0002-9947-1977-0439876-2 |
| Reference: | [19] Nicholson, W. K., Zhou, Y.: Rings in which elements are uniquely the sum of an idempotent and a unit.Glasg. Math. J. 46 (2004), 227-236. Zbl 1057.16007, MR 2062606, 10.1017/S0017089504001727 |
| Reference: | [20] Passman, D. S.: The Algebraic Structure of Group Rings.Dover, New York (2011). MR 0470211 |
| Reference: | [21] Sabet, S. A. S., Farmani, M., Karami, M.: $\pi$-clean and strongly $\pi$-clean rings.Measurement 109 (2017), 74-78. 10.1016/j.measurement.2017.05.037 |
| Reference: | [22] Sahinkaya, S., Tang, G., Zhou, Y.: Nil-clean group rings.J. Algebra Appl. 16 (2017), Article ID 1750135, 7 pages. Zbl 1382.16021, MR 3660418, 10.1142/S0219498817501353 |
| Reference: | [23] Wang, X., You, H.: Cleanness of the group ring of an Abelian $p$-group over a commutative ring.Algebra Colloq. 19 (2012), 539-544. Zbl 1259.16026, MR 2999263, 10.1142/S1005386712000405 |
| Reference: | [24] Zhou, Y.: On clean group rings.Advances in Ring Theory Trends in Mathematics. Birkhäuser, Basel (2010), 335-345. Zbl 1205.16020, MR 2664681, 10.1007/978-3-0346-0286-0_22 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
