It is observed that unit group of these rings behave differently from the unit groups of $M_2(\mathbb{Z}_2)$ and $M_2(\mathbb{Z}_3)$.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:The aim of this paper is to study idempotents and units in certain matrix rings over polynomial rings.
📊
Key Findings & Data:More precisely, the conditions under which an element in $M_2(\mathbb{Z}_p[x])$ for any prime $p$, an element in $M_2(\mathbb{Z}_{2p}[x])$ for any odd prime $p$, and an element in $M_2(\mathbb{Z}_{3p}[x])$ for any prime $p$ greater than 3 is an idempotent are obtained and these conditions are used to give the form of idempotents in these matrix rings.
💡
Conclusion & Impact:It is observed that unit group of these rings behave differently from the unit groups of $M_2(\mathbb{Z}_2)$ and $M_2(\mathbb{Z}_3)$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Idempotent · unit
Abstract
The aim of this paper is to study idempotents and units in certain matrix rings over polynomial rings. More precisely, the conditions under which an element in $M_2(\mathbb{Z}_p[x])$ for any prime $p$, an element in $M_2(\mathbb{Z}_{2p}[x])$ for any odd prime $p$, and an element in $M_2(\mathbb{Z}_{3p}[x])$ for any prime $p$ greater than 3 is an idempotent are obtained and these conditions are used to give the form of idempotents in these matrix rings. The form of elements in $M_2(\mathbb{Z}_2[x])$ and elements in $M_2(\mathbb{Z}_3[x])$ that are units is also given. It is observed that unit group of these rings behave differently from the unit groups of $M_2(\mathbb{Z}_2)$ and $M_2(\mathbb{Z}_3)$.
KANWAR, P., KHATKAR, M., SHARMA, R. K. (2017). Idempotents and Units of Matrix Rings over Polynomial Rings. International Electronic Journal of Algebra, Cilt: 22 - Sayı: 22, 147–169. https://ajindex.com/articles/832352
P. B. Bhattacharya and S. K. Jain, A note on the adjoint group of a ring, Arch. Math. (Basel), 21 (1970), 366-368.
P. B. Bhattacharya and S. K. Jain, Rings having solvable adjoint groups, Proc. Amer. Math. Soc., 25 (1970), 563-565.
V. Bovdi and M. Salim, On the unit group of a commutative group ring, Acta Sci. Math. (Szeged), 80(3-4) (2014), 433-445.
D. M. Burton, Elementary Number Theory, McGraw-Hill Education, 7th edi- tion, 2011.
J. L. Fisher, M. M. Parmenter and S. K. Sehgal, Group rings with solvable n-Engel unit groups, Proc. Amer. Math. Soc., 59(2) (1976), 195-200.
K. R. Goodearl, Von Neumann Regular Rings, Second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.
P. Kanwar, A. Leroy and J. Matczuk, Idempotents in ring extensions, J. Alge- bra, 389 (2013), 128-136.
P. Kanwar, A. Leroy and J. Matczuk, Clean elements in polynomial rings, Noncommutative Rings and their Applications, Contemp. Math., Amer. Math. Soc., 634 (2015), 197-204.
P. Kanwar, R. K. Sharma and P. Yadav, Lie regular generators of general linear groups II, Int. Electron. J. Algebra, 13 (2013), 91-108.
C. Lanski, Some remarks on rings with solvable units, Ring Theory, (Proc. Conf., Park City, Utah, 1971),
Academic Press, New York, (1972), 235-240.
B. R. McDonald, Linear Algebra over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, 87, Marcel Dekker, Inc., New York, 1984.
M. Mirowicz, Units in group rings of the infinite dihedral group, Canad. Math. Bull., 34(1) (1991), 83-89.
W. K. Nicholson, 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 and P. Kanwar, Lie regular generators of general linear groups, Comm. Algebra, 40(4) (2012), 1304-1315.
L. Wiechecki, Group algebra units and tree actions, J. Algebra, 311(2) (2007), 781-799.