Involutives in Triangular Matrix Rings

Yıl 2020, Cilt: 7 Sayı: 1, 91 - 103, 28.06.2020

Tuğba Petik , Leman Hocaoğlu , Halim Özdemir

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

Öz

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. 

Anahtar Kelimeler

Involutive matrices, Triangular matrices, Matrix rings

Üçgensel Matris Halkalarında İnvolutifler

Öz

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.

Anahtar Kelimeler

İnvolutif matrisler, Üçgensel matrisler, Matris halkaları

Kaynakça

• 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.