IDEMPOTENTS IN CERTAIN MATRIX RINGS OVER POLYNOMIAL RINGS

Yıl 2020, , 1 - 12, 07.01.2020

Jose Maria P. Balmaceda Joanne Pauline P. Datu

https://doi.org/10.24330/ieja.662942

Cited By: 1

Öz

We determine the forms of the nontrivial idempotents in the ring of $2\times 2$ matrices over the polynomial rings $\mathbb{Z}_{pq}[x]$ and $\mathbb{Z}_{p^2}[x]$, where $p$ and $q$ are any primes. Any such idempotent in the stated rings will be of a form in our list. Our work generalizes the results of Kanwar, Khatkar and Sharma (2017) who identified the forms of idempotents in $M_2(\mathbb{Z}_{2p}[x])$ and $M_2(\mathbb{Z}_{3p}[x])$.

Anahtar Kelimeler

Idempotent, matrix ring, polynomial ring

Kaynakça

• P. N. Anh, G. F. Birkenmeier and L. van Wyk, Idempotents and structures of rings, Linear Multilinear Algebra, 64(10) (2016), 2002-2029.
• D. M. Burton, Elementary Number Theory, 6th Edition, Tata McGraw-Hill Education Pvt. Ltd., 2006.
• M. Henriksen, Two classes of rings generated by their units, J. Algebra, 31 (1974), 182-193.
• T. W. Hungerford, Abstract Algebra: An Introduction, 3rd Edition, Cengage Learning, 2012.
• P. Kanwar, M. Khatkar and R. K. Sharma, Idempotents and units of matrix rings over polynomial rings, Int. Electron. J. Algebra, 22 (2017), 147-169.
• P. Kanwar, A. Leroy and J. Matczuk, Idempotents in ring extensions, J. Alge- bra, 389 (2013), 128-136.
• E. D. Nering, Linear Algebra and Matrix Theory, 2nd Edition, John Wiley & Sons Inc., 1970.
• W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasg. Math. J., 46(2) (2004), 227-236.
• A. K. Srivastava, Additive representations of elements in rings: a survey, in Algebra and its Applications, Springer Proc. Math. Stat., Springer, Singapore, 174 (2016), 59-73.