Research Article
Year 2017,
Volume: 22 Issue: 22, 147 - 169, 11.07.2017
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 $M_2(\mathbb{Z}_3)$.
Keywords
References
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Year 2017,
Volume: 22 Issue: 22, 147 - 169, 11.07.2017
Abstract
References
- 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.
There are 16 citations in total.
Details
| Subjects | Mathematical Sciences |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | July 11, 2017 |
| Published in Issue | Year 2017 Volume: 22 Issue: 22 |