The aim of the present paper is to characterize associative rings R with unity in which 1+eR(1-e) in terms of some important class of rings in the literature (for example, NR-rings, UU-rings, UJ-rings, UR-rings, exchange rings, 2-primal rings), where $e^2=e\in\ R$ and $U(R)$ is the set of units of R.
[1] Chun Y, Jeon, YC, Kang S, Lee, KN. A concept unifying the Armendariz and NI conditions. Bull Korean Math Soc 2011; 48 (1): 115-127.
[2] Cohn PM. Rings of zero divisors. Proc Amer Math Soc 1958; 9: 914-919.
[3] Henriksen M. Rings with a unique regular element. Rings, modules and radicals. Pitman Res Notes Math Ser 204, Harlow England: Longman Sci Tech, 1989. pp. 78-87.
[4] Hwang SU and Jeon YC. Structure and topological conditions of NI rings, J Algebra 2006; 302 (1): 186-199.
[5] Koşan MT, Leroy A, Matczuk J. On UJ-rings. Commun Algebra 2018; 46: 2297-2303.
[6] Koşan MT, Quynh TC, Yildirim T, \breve{Z}emli\breve{c}ka J. Rings such that, for each unit u, u-u^n belongs to the Jacobson radical. Hacettepe J Math 2020; 49 (4): 1397 – 1404.
[7] Lam TY. Rings with unipotent units. Publ Math Debrecen 2016; 88: 449-466.
[8] Nicholson WK. Lifting idempotents and exchange rings. Trans Amer Math Soc 1977; 229: 269-278.
[9] Porter T. Cohn’s rings of zero divisors. Arch Math 1984; 43: 340-343.
[10] Rosenberg A. The structure of the infinite general linear group. Ann of Math 1958; 68: 278-293.
[11] Sahinkaya S, Yildirim T. UJ-endomorphism rings. The Mathematica Journal 2018; 60 (83) (2): 186-198.
[12] Ster J. Rings in which nilpotents form a subring. Carpathian Journal of Mathematics 2016; 32 (2): 251-258.
ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e)
The aim of the present paper is to characterize associative rings R with unity in which 1+eR(1-e) in terms of some important class of rings in the literature (for example, NR-rings, UU-rings, UJ-rings, UR-rings, exchange rings, 2-primal rings), where e2=e∈ R and U(R) is the set of units of R.
[1] Chun Y, Jeon, YC, Kang S, Lee, KN. A concept unifying the Armendariz and NI conditions. Bull Korean Math Soc 2011; 48 (1): 115-127.
[2] Cohn PM. Rings of zero divisors. Proc Amer Math Soc 1958; 9: 914-919.
[3] Henriksen M. Rings with a unique regular element. Rings, modules and radicals. Pitman Res Notes Math Ser 204, Harlow England: Longman Sci Tech, 1989. pp. 78-87.
[4] Hwang SU and Jeon YC. Structure and topological conditions of NI rings, J Algebra 2006; 302 (1): 186-199.
[5] Koşan MT, Leroy A, Matczuk J. On UJ-rings. Commun Algebra 2018; 46: 2297-2303.
[6] Koşan MT, Quynh TC, Yildirim T, \breve{Z}emli\breve{c}ka J. Rings such that, for each unit u, u-u^n belongs to the Jacobson radical. Hacettepe J Math 2020; 49 (4): 1397 – 1404.
[7] Lam TY. Rings with unipotent units. Publ Math Debrecen 2016; 88: 449-466.
[8] Nicholson WK. Lifting idempotents and exchange rings. Trans Amer Math Soc 1977; 229: 269-278.
[9] Porter T. Cohn’s rings of zero divisors. Arch Math 1984; 43: 340-343.
[10] Rosenberg A. The structure of the infinite general linear group. Ann of Math 1958; 68: 278-293.
[11] Sahinkaya S, Yildirim T. UJ-endomorphism rings. The Mathematica Journal 2018; 60 (83) (2): 186-198.
[12] Ster J. Rings in which nilpotents form a subring. Carpathian Journal of Mathematics 2016; 32 (2): 251-258.
Özdin, T. (2022). ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e). Eskişehir Teknik Üniversitesi Bilim Ve Teknoloji Dergisi B - Teorik Bilimler, 10(1), 11-17. https://doi.org/10.20290/estubtdb.957366
AMA
Özdin T. ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e). Estuscience - Theory. Şubat 2022;10(1):11-17. doi:10.20290/estubtdb.957366
Chicago
Özdin, Tufan. “ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-E)”. Eskişehir Teknik Üniversitesi Bilim Ve Teknoloji Dergisi B - Teorik Bilimler 10, sy. 1 (Şubat 2022): 11-17. https://doi.org/10.20290/estubtdb.957366.
EndNote
Özdin T (01 Şubat 2022) ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e). Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 10 1 11–17.
IEEE
T. Özdin, “ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e)”, Estuscience - Theory, c. 10, sy. 1, ss. 11–17, 2022, doi: 10.20290/estubtdb.957366.
ISNAD
Özdin, Tufan. “ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-E)”. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 10/1 (Şubat 2022), 11-17. https://doi.org/10.20290/estubtdb.957366.
JAMA
Özdin T. ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e). Estuscience - Theory. 2022;10:11–17.
MLA
Özdin, Tufan. “ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-E)”. Eskişehir Teknik Üniversitesi Bilim Ve Teknoloji Dergisi B - Teorik Bilimler, c. 10, sy. 1, 2022, ss. 11-17, doi:10.20290/estubtdb.957366.
Vancouver
Özdin T. ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e). Estuscience - Theory. 2022;10(1):11-7.