Involutives in Triangular Matrix Rings

Year 2020, Volume: 7 Issue: 1, 91 - 103, 28.06.2020

Tuğba Petik , Leman Hocaoğlu , Halim Özdemir

https://doi.org/10.35193/bseufbd.687316

Abstract

Let R be a commutative ring with identity 1 whose involutives are only -1 and 1, and let M be an upper triangular matrices ring which entries are taken from the ring R. In the study, it is established the necessary and sufficient conditions for an element taken from the ring M  to be involutive. Also, when R is finite, it is given a result determining the number of involutive elements in the ring M, and this result is supported by numerical examples. 

Keywords

Involutive matrices , Triangular matrices , Matrix rings

Üçgensel Matris Halkalarında İnvolutifler

Abstract

R, birimli, involutifleri sadece -1 ve 1 olan bir değişmeli halka ve M, elemanları R halkası üzerinden alınan bir üst üçgensel matrisler halkası olsun. Çalışmada, M halkasından alınan bir elemanın involutif olması için gerek ve yeter koşullar ortaya koyulmaktadır. Ayrıca, R sonlu olduğunda, M halkasındaki involutif elemanların sayısını belirleyen bir sonuç verilmekte ve bu sonuç sayısal örneklerle desteklenmektedir.

Keywords

İnvolutif matrisler , Üçgensel matrisler , Matris halkaları

References

• Hannah J., O’Meara K.C. (1991). Products of Simultaneously Triangulable Idempotent Matrices. Linear Algebra and its Applications, 149, 185-190.
• Fošner, A. (2005). Automorphisms of the poset of upper triangular idempotent matrices. Linear and Multilinear Algebra, 53(1), 27-44.
• Chen, J., Wang, Z., Zhou, Y. (2009). Rings in which elements are uniquely the sum of an idempotent and a unit that commute. Journal of Pure and Applied Algebra, 213, 215–223.
• Ying, Z., Koşan, T., and Zhou,Y. (2016). Rings in which Every Element is a Sum of Two Tripotents. Canad. Math. Bull., 59(3), 661–672.
• Sheibani, M. and Huanyin, C. (2017). Rings over which every matrix is the sum of a tripotent and a nilpotent. https://arxiv.org/abs/1702.05605
• Zhou, Y. (2018). Rings in which elements are sums of nilpotents, idempotents and tripotents. Journal of Algebra and Its Applications, 17(1), 1850009 (7 pages).
• Danchev, P.V. (2018). Rings whose elements are sums of three or minus sums of two commuting idempotents, Albanian Journal of Mathematics, 12(1), 3–7.
• Cheraghpour, H. and Ghosseiri, Nader M. (2019). On the idempotents, nilpotents, units and zerodivisors of finite rings, Linear and Multilinear Algebra, 67(2), 327–336.
• Tang, G., Zhou, Y., and Su, H. (2019). Matrices over a commutative ring as sums of three idempotents or three involutions, Linear and Multilinear Algebra, 67(2), 267–277.
• Hou, X. Idempotents in Triangular Matrix Rings, Linear and Multilinear Algebra, https://doi.org/10.1080/03081087.2019.1596223
• Hirano, Y. and Tominaga, H. (1988). Rings in which every element is the sum of two idempotents. Bull. Austral. Math. Soc., 37(2), 161-164.
• de Seguins C. Pazzis. (2010). On sums of idempotent matrices over a field of positive characteristic, Linear Algebra Appl., 433(4), 856–866.