Research Article
BibTex RIS Cite

Idempotents and Units of Matrix Rings over Polynomial Rings

Year 2017, , 147 - 169, 11.07.2017
https://doi.org/10.24330/ieja.325941

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 $M_2(\mathbb{Z}_3)$.
 

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.
Year 2017, , 147 - 169, 11.07.2017
https://doi.org/10.24330/ieja.325941

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

Pramod Kanwar

Meenu Khatkar This is me

R. K. Sharma This is me

Publication Date July 11, 2017
Published in Issue Year 2017

Cite

APA Kanwar, P., Khatkar, M., & Sharma, R. K. (2017). Idempotents and Units of Matrix Rings over Polynomial Rings. International Electronic Journal of Algebra, 22(22), 147-169. https://doi.org/10.24330/ieja.325941
AMA Kanwar P, Khatkar M, Sharma RK. Idempotents and Units of Matrix Rings over Polynomial Rings. IEJA. July 2017;22(22):147-169. doi:10.24330/ieja.325941
Chicago Kanwar, Pramod, Meenu Khatkar, and R. K. Sharma. “Idempotents and Units of Matrix Rings over Polynomial Rings”. International Electronic Journal of Algebra 22, no. 22 (July 2017): 147-69. https://doi.org/10.24330/ieja.325941.
EndNote Kanwar P, Khatkar M, Sharma RK (July 1, 2017) Idempotents and Units of Matrix Rings over Polynomial Rings. International Electronic Journal of Algebra 22 22 147–169.
IEEE P. Kanwar, M. Khatkar, and R. K. Sharma, “Idempotents and Units of Matrix Rings over Polynomial Rings”, IEJA, vol. 22, no. 22, pp. 147–169, 2017, doi: 10.24330/ieja.325941.
ISNAD Kanwar, Pramod et al. “Idempotents and Units of Matrix Rings over Polynomial Rings”. International Electronic Journal of Algebra 22/22 (July 2017), 147-169. https://doi.org/10.24330/ieja.325941.
JAMA Kanwar P, Khatkar M, Sharma RK. Idempotents and Units of Matrix Rings over Polynomial Rings. IEJA. 2017;22:147–169.
MLA Kanwar, Pramod et al. “Idempotents and Units of Matrix Rings over Polynomial Rings”. International Electronic Journal of Algebra, vol. 22, no. 22, 2017, pp. 147-69, doi:10.24330/ieja.325941.
Vancouver Kanwar P, Khatkar M, Sharma RK. Idempotents and Units of Matrix Rings over Polynomial Rings. IEJA. 2017;22(22):147-69.