Mathematica Bohemica, first online, pp. 1-13


On $\mathcal Z$-reflexive rings

Nirbhay Kumar

Received December 15, 2024.   Published online March 24, 2025.

Abstract:  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.
Keywords:  reflexive ring; $\mathcal Z$-reflexive ring; $J$-reflexive ring; central reflexive ring; singular ideal
Classification MSC:  13C99, 16U99, 16N20, 16D80

PDF available at:  Institute of Mathematics CAS

References:
[1] A. Amini, B. Amini, A. Nejadzadeh, H. Sharif: Singular clean rings. J. Korean Math. Soc. 55 (2018), 1143-1156. DOI 10.4134/JKMS.j170615 | MR 3849355 | Zbl 1396.16032
[2] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13. Springer, New York (1974). DOI 10.1007/978-1-4612-4418-9 | MR 0417223 | Zbl 0765.16001
[3] M. B. Çalci, H. Chen, S. Halicioğlu: A generalization of reflexive rings. Math. Bohem. 149 (2024), 225-235. DOI 10.21136/MB.2023.0034-22 | MR 4767009 | Zbl 07893420
[4] M. B. Calci, H. Chen, S. Halicioglu, A. Harmanci: Reversibility of rings with respect to the Jacobson radical. Mediterr. J. Math. 14 (2017), Article ID 137, 14 pages. DOI 10.1007/s00009-017-0938-2 | MR 3654896 | Zbl 1377.16034
[5] U. S. Chakraborty: On some classes of reflexive rings. Asian-Eur. J. Math. 8 (2015), Article ID 1550003, 15 pages. DOI 10.1142/S1793557115500035 | MR 3322546 | Zbl 1327.16024
[6] A. K. Chaturvedi, N. Kumar: On $z$-reversible rings. Proc. Natl. Acad. Sci. India, Sect. A, Phys. Sci. 92 (2022), 555-562. DOI 10.1007/s40010-022-00770-3 | MR 4515911 | Zbl 1515.16035
[7] P. M. Cohn: Reversible rings. Bull. Lond. Math. Soc. 31 (1999), 641-648. DOI 10.1112/S0024609399006116 | MR 1711020 | Zbl 1021.16019
[8] H. Kose, B. Ungor, S. Halicioglu, A. Harmanci: A generalization of reversible rings. Iran. J. Sci. Technol., Trans. A, Sci. 38 (2014), 43-48. DOI 10.22099/ijsts.2014.1903 | MR 3288572
[9] T. Y. Lam: Lectures on Modules and Rings. Graduate Texts in Mathematics 189. Springer, New York (1999). DOI 10.1007/978-1-4612-0525-8 | MR 1653294 | Zbl 0911.16001
[10] T. Y. Lam: Exercises in Modules and Rings. Problem Books in Mathematics. Springer, New York (2007). DOI 10.1007/978-0-387-48899-8 | MR 2278849 | Zbl 1121.16001
[11] G. Mason: Reflexive ideals. Commun. Algebra 9 (1981), 1709-1724. DOI 10.1080/00927878108822678 | MR 0631884 | Zbl 0468.16024
[12] A. Nejadzadeh, A. Amini, B. Amini, H. Sharif: Rings in which nilpotent elements are right singular. Bull. Iran. Math. Soc. 44 (2018), 1217-1226. DOI 10.1007/s41980-018-0085-y | MR 3861469 | Zbl 1407.16036
[13] W. K. Nicholson: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229 (1977), 269-278. DOI 10.1090/S0002-9947-1977-0439876-2 | MR 0439876 | Zbl 0352.16006
[14] M. B. Rege, S. Chhawchharia: Armendariz rings. Proc. Japan Acad., Ser. A 73 (1997), 14-17. DOI 10.3792/pjaa.73.14 | MR 1442245 | Zbl 0960.16038
[15] R. Yue Chi Ming: On quasi-Frobeniusean and Artinian rings. Publ. Inst. Math., Nouv. Sér. 33 (1983), 239-245. MR 0723453 | Zbl 0521.16009

Affiliations:   Nirbhay Kumar, Feroze Gandhi College, District Collectorate, Kutchery Road, Raebareli-229001, India, e-mail: nirbhayk2897@gmail.com


 
PDF available at: