| Title:
|
On $\mathcal Z$-reflexive rings (English) |
| Author:
|
Kumar, Nirbhay |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
2 |
| Year:
|
2026 |
| Pages:
|
291-303 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce the notion of $\mathcal Z$-reflexive rings to describe reflexivity of rings in terms of their singular ideals. We show that $\mathcal Z$-reflexive ring is proper common generalization of a central reflexive ring, $\mathcal Z$-reversible ring, and singular clean ring. We discuss some its properties, characterizations, and relations with some extension rings. We show that a ring $R$ is right $\mathcal Z$-reflexive if and only if $M_n(R)$ is right $\mathcal Z$-reflexive for every positive integer $n$. Also, we share the connection of right $\mathcal Z$-reflexive rings with $J$-reflexive rings. (English) |
| Keyword:
|
reflexive ring |
| Keyword:
|
$\mathcal Z$-reflexive ring |
| Keyword:
|
$J$-reflexive ring |
| Keyword:
|
central reflexive ring |
| Keyword:
|
singular ideal |
| MSC:
|
13C99 |
| MSC:
|
16D80 |
| MSC:
|
16N20 |
| MSC:
|
16U99 |
| DOI:
|
10.21136/MB.2025.0163-24 |
| . |
| Date available:
|
2026-05-19T08:23:53Z |
| Last updated:
|
2026-05-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153625 |
| . |
| Reference:
|
[1] 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:
|
[2] Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules.Graduate Texts in Mathematics 13. Springer, New York (1974). Zbl 0765.16001, MR 0417223, 10.1007/978-1-4612-4418-9 |
| Reference:
|
[3] Çalcı, M. B., Chen, H., oğlu, S. Halıcı: A generalization of reflexive rings.Math. Bohem. 149 (2024), 225-235. Zbl 07893420, MR 4767009, 10.21136/MB.2023.0034-22 |
| Reference:
|
[4] Calci, M. B., Chen, H., Halicioglu, S., Harmanci, A.: Reversibility of rings with respect to the Jacobson radical.Mediterr. J. Math. 14 (2017), Article ID 137, 14 pages. Zbl 1377.16034, MR 3654896, 10.1007/s00009-017-0938-2 |
| Reference:
|
[5] Chakraborty, U. S.: On some classes of reflexive rings.Asian-Eur. J. Math. 8 (2015), Article ID 1550003, 15 pages. Zbl 1327.16024, MR 3322546, 10.1142/S1793557115500035 |
| Reference:
|
[6] Chaturvedi, A. K., Kumar, N.: On $z$-reversible rings.Proc. Natl. Acad. Sci. India, Sect. A, Phys. Sci. 92 (2022), 555-562. Zbl 1515.16035, MR 4515911, 10.1007/s40010-022-00770-3 |
| Reference:
|
[7] Cohn, P. M.: Reversible rings.Bull. Lond. Math. Soc. 31 (1999), 641-648. Zbl 1021.16019, MR 1711020, 10.1112/S0024609399006116 |
| Reference:
|
[8] Kose, H., Ungor, B., Halicioglu, S., Harmanci, A.: A generalization of reversible rings.Iran. J. Sci. Technol., Trans. A, Sci. 38 (2014), 43-48. MR 3288572, 10.22099/ijsts.2014.1903 |
| Reference:
|
[9] Lam, T. Y.: Lectures on Modules and Rings.Graduate Texts in Mathematics 189. Springer, New York (1999). Zbl 0911.16001, MR 1653294, 10.1007/978-1-4612-0525-8 |
| Reference:
|
[10] Lam, T. Y.: Exercises in Modules and Rings.Problem Books in Mathematics. Springer, New York (2007). Zbl 1121.16001, MR 2278849, 10.1007/978-0-387-48899-8 |
| Reference:
|
[11] Mason, G.: Reflexive ideals.Commun. Algebra 9 (1981), 1709-1724. Zbl 0468.16024, MR 0631884, 10.1080/00927878108822678 |
| Reference:
|
[12] Nejadzadeh, A., Amini, A., Amini, B., Sharif, H.: Rings in which nilpotent elements are right singular.Bull. Iran. Math. Soc. 44 (2018), 1217-1226. Zbl 1407.16036, MR 3861469, 10.1007/s41980-018-0085-y |
| Reference:
|
[13] Nicholson, W. K.: Lifting idempotents and exchange rings.Trans. Am. Math. Soc. 229 (1977), 269-278. Zbl 0352.16006, MR 0439876, 10.1090/S0002-9947-1977-0439876-2 |
| Reference:
|
[14] Rege, M. B., Chhawchharia, S.: Armendariz rings.Proc. Japan Acad., Ser. A 73 (1997), 14-17. Zbl 0960.16038, MR 1442245, 10.3792/pjaa.73.14 |
| Reference:
|
[15] Ming, R. Yue Chi: On quasi-Frobeniusean and Artinian rings.Publ. Inst. Math., Nouv. Sér. 33 (1983), 239-245. Zbl 0521.16009, MR 0723453 |
| . |
