Reversibility of skew Hurwitz series rings

Year 2020, Volume: 49 Issue: 6, 2074 - 2083, 08.12.2020

Fatma Kaynarca , Muhammed Ali Yıldırım

https://doi.org/10.15672/hujms.679606

Cited By: 1

Abstract

We study the reversibility of skew Hurwitz series at zero as a generalization of an $\alpha$-rigid ring, introducing the concept of skew Hurwitz reversibility. A ring $R$ is called skew Hurwitz reversible ($SH$-reversible, for short), if the skew Hurwitz series ring $(HR,\alpha)$ is reversible i.e. whenever skew Hurwitz series $f, g\in (HR,\alpha)$ satisfy $fg=0$, then $gf=0$. We examine some characterizations and extensions of $SH$-reversible rings in relation with several ring theoretic properties which have roles in ring theory.

Keywords

Skew Hurwitz series ring, reversible ring, α-rigid ring, matrix ring

References

• [1] M. Ahmadi, A. Moussavi and V. Nourozi, On skew Hurwitz serieswise Armendariz rings, Asian-Eur. J. Math. 7 (3), 1450036, 2014.
• [2] D.D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra, 27 (6), 2847–2852, 1999.
• [3] M. Başer, C.Y. Hong and T.K. Kwak, On Extended Reversible Rings, Algebra Colloq. 16 (1), 37–48, 2009.
• [4] A. Benhissi and F. Koja, Basic propoerties of Hurwitz series rings, Ric. Mat. 61, 255–273, 2012.
• [5] P.M. Cohn, Reversible rings, Bull. Lond. Math. Soc. 31, 641–648, 1999.
• [6] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 3, 207–224, 2005.
• [7] A.M Hassanein, Clean rings of skew hurwitz series, Matematiche, 62 (1), 47–54, 2007.
• [8] A.M. Hassanein, On uniquely clean skew Hurwitz series, Southeast Asian Bull. Math. 35, 5-10, 2012.
• [9] C.Y. Hong, N.K. Kim and T.K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra, 151, 215–226, 2000.
• [10] C.Y. Hong, N.K. Kim and T.K. Kwak, On skew Armendariz rings, Comm. Algebra, 31, 103–122, 2003.
• [11] C.Y. Hong, T.K. Kwak and S.T. Rizvi, Extensions of generalized Armendariz rings, Algebra Colloq. 13, 253–266, 2006.
• [12] H.L. Jin, F. Kaynarca, T.K. Kwak and Y. Lee, On commutativity of skew polynomials at zero, Bull. Korean Math. Soc. 54, 51–69, 2017.
• [13] W.F. Keigher, Adjunctions and comonads in differential algebra, Pacific J. Math. 59, 99–112, 1975.
• [14] W.F. Keigher, On the ring of Hurwitz Series, Comm. Algebra, 25 (6), 1845–1859, 1997.
• [15] W.F. Keigher and F.L. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra, 146, 291-304, 2000.
• [16] N.K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra, 185, 207–223, 2003.
• [17] J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (4), 289–300, 1996.
• [18] J. Krempa and D. Niewieczerzal, Rings in which annihilators are ideals and their application to semigroup rings, Bull. Acad. Polon. Sci. Ser. Sci., Math. Astronom. Phys, 25, 851-856, 1977.
• [19] J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (3), 359-368, 1971.
• [20] K. Paykan, A study on skew Hurwitz series ring, Ric. Mat. 66 (2), 383–393, 2016.
• [21] K. Paykan, Principally quasi-Baer skew Hurwitz series rings, Boll. Unione Mat. Ital. 10 (4), 607–616, 2017.
• [22] M.B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73, 14–17, 1997.
• [23] S. Veldsman, Matrix and polynomial reversibility of rings, Comm. Algebra, 43, 1571– 1582, 2015.
• [24] G. Yang and Z.K. Liu, On strongly reversible rings, Taiwanese J. Math. 12 (1), 129– 136, 2008.