Yıl 2017, Cilt 22, Sayı 22, Sayfalar 147 - 169 2017-07-11
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## Idempotents and Units of Matrix Rings over Polynomial Rings

#### Pramod Kanwar [1] , Meenu Khatkar [2] , R. K. Sharma [3]

##### 151 91

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 $M_2(\mathbb{Z}_3)$.

Idempotent, unit
• 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.
Konular Matematik Makaleler Yazar: Pramod Kanwar Yazar: Meenu Khatkar Yazar: R. K. Sharma
 Bibtex @araştırma makalesi { ieja325941, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2017}, volume = {22}, pages = {147 - 169}, doi = {10.24330/ieja.325941}, title = {Idempotents and Units of Matrix Rings over Polynomial Rings}, key = {cite}, author = {Sharma, R. K. and Kanwar, Pramod and Khatkar, Meenu} } APA Kanwar, P , Khatkar, M , Sharma, R . (2017). Idempotents and Units of Matrix Rings over Polynomial Rings. International Electronic Journal of Algebra, 22 (22), 147-169. DOI: 10.24330/ieja.325941 MLA Kanwar, P , Khatkar, M , Sharma, R . "Idempotents and Units of Matrix Rings over Polynomial Rings". International Electronic Journal of Algebra 22 (2017): 147-169 Chicago Kanwar, P , Khatkar, M , Sharma, R . "Idempotents and Units of Matrix Rings over Polynomial Rings". International Electronic Journal of Algebra 22 (2017): 147-169 RIS TY - JOUR T1 - Idempotents and Units of Matrix Rings over Polynomial Rings AU - Pramod Kanwar , Meenu Khatkar , R. K. Sharma Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.325941 DO - 10.24330/ieja.325941 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 147 EP - 169 VL - 22 IS - 22 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.325941 UR - http://dx.doi.org/10.24330/ieja.325941 Y2 - 2018 ER - EndNote %0 International Electronic Journal of Algebra Idempotents and Units of Matrix Rings over Polynomial Rings %A Pramod Kanwar , Meenu Khatkar , R. K. Sharma %T Idempotents and Units of Matrix Rings over Polynomial Rings %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 22 %N 22 %R doi: 10.24330/ieja.325941 %U 10.24330/ieja.325941 ISNAD Kanwar, Pramod , Khatkar, Meenu , Sharma, R. K. . "Idempotents and Units of Matrix Rings over Polynomial Rings". International Electronic Journal of Algebra 22 / 22 (Temmuz 2017): 147-169. http://dx.doi.org/10.24330/ieja.325941