Non-trivial idempotents of the matrix rings over the polynomial ring \(Z_{pqr}[x]\)
Abstract
In this paper, we study the non-trivial idempotents of the $2 \times 2$ matrix ring over the polynomial ring \(\mathbb{Z}_{pqr}[x]\) for distinct primes \(p, q\) and \(r\) greater than 3. We have classified all the idempotents of this matrix ring into several classes such that any idempotent must belong to one of these classes. This work is extension of the work done in [1].
2020 Mathematics Subject Classification:
16S50, 13F20Keywords
idempotent, polynomial ring, matrix ring
Author Details
Gaurav Mittal
Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee, India
email: gmittal@ma.iitr.ac.in
References
- J. M. P. Balmaceda, J. P. P. Datu. Idempotents in certain matrix rings over polynomial rings. Int. Electron. J. Algebra 27 (2020), 1–12.
- K. R. Goodearl. Von Neumann Regular Rings, 2nd ed. Malabar, Fl, Krieger Pub. Co., 1991.
- P. Kanwar, M. Khatkar, R. K. Sharma. Idempotents and units of matrix rings over polynomial rings. Int. Electron. J. Algebra 22 (2017), 147–169.
- P. Kanwar, A. Leroy, J. Matczuk. Idempotents in ring extensions. J. Algebra 389 (2013), 128–136.
- P. Kanwar, A. Leroy, J. Matczuk. Clean elements in polynomial rings. Contemp. Math. 634 (2015), 197–204.
- W. K. Nicholson. Lie Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229 (1977), 269–278.
- W. K. Nicholson. Strongly clean rings and Fitting’s lemma. Comm. Algebra, 27, 8 (1999), 3583–3592.
- R. K. Sharma, P. Yadav, P. Kanwar. Lie regular generators of general linear groups. Comm. Algebra, 40, 4 (2012), 1304–1315.