Jordan regular units in rings and group rings

P.  Kumari; M.  Sahai; R. K.  Sharma

doi:10.37863/umzh.v75i3.1130

P. Kumari Govt. P. G. College for Women, Rohtak, Haryana, India https://orcid.org/0000-0001-9178-1532
M. Sahai Department of Mathematics and Astronomy, Lucknow University, India https://orcid.org/0000-0002-2709-9210
R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, India

DOI: https://doi.org/10.37863/umzh.v75i3.1130

Keywords: JORDAN REGULAR UNITS

Abstract

UDC 512.5

The concept of Lie regular elements and Lie regular units was defined and studied by Kanwar, Sharma and Yadav in Lie regular generators of general linear groups, Comm. Algebra, 40, № 4, 1304–1315 (2012)]. We introduce Jordan regular elements and Jordan regular units. It is proved that the order of the set of Jordan regular units in $M(2, Z_{2^n})$ is equal to a half of the order of $U(M(2,Z_{2^n})).$ Further, we show that the group ring $KG$ of a group $G$ over a field $K$ of characteristic $2$ has no Jordan regular units.

References

P. Kanwar, R. K. Sharma, P. Yadav, Lie regular generators of general linear groups, Comm. Algebra, 40, № 4, 1304–1315 (2012). DOI: https://doi.org/10.1080/00927872.2010.551526

S. Maheshwari, R. K. Sharma, Lie regular generators of general linear group $GL(4, Z_n)$, Serdica Math. J., 42, № 3-4, 211–220 (2016).

N. Makhijani, R. K. Sharma, J. B. Srivastava, The unit group of $F_q[D_{30}]$, Serdica Math. J., 41, 185–198 (2015).

N. Makhijani, R. K. Sharma, J. B. Srivastava, On the order of unitary subgroup of the modular group algebra $F_{2^k}D_{2N}$, J. Algebra and Appl., 14, № 8, Article 1550129 (2015). DOI: https://doi.org/10.1142/S0219498815501297

M. Sahai, R. K. Sharma, P. Kumari, Jordan regular generators of general linear groups, J. Indian Math. Soc., 85, № 3-4, 422–433 (2018). DOI: https://doi.org/10.18311/jims/2018/20971

R. K. Sharma, S. Gangopadhyay, V. Vetrivel, On units in $ZS_3$, Comm. Algebra, 25, 2285–2299 (1997). DOI: https://doi.org/10.1080/00927879708825989

R. K. Sharma, J. B. Srivastava, M. Khan, The unit group of $FA_4$, Publ. Math. Debrecen, 71, № 1-2, 21–26 (2007). DOI: https://doi.org/10.5486/PMD.2007.3541

R. K. Sharma, J. B. Srivastava, M. Khan, The unit group of $FS_4$, Acta Math. Hungar., 118, № 1-2, 105–113 (2008). DOI: https://doi.org/10.1007/s10474-007-6169-4

R. K. Sharma, P. Yadav, The unit group of $Z_pQ_8$, Algebras, Groups and Geom., 25, 425–430 (2008).