Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-11T09:06:50.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings in which every element is the sum of two idempotents

Published online by Cambridge University Press:  17 April 2009

Yasuyuki Hirano  and
Hisao Tominaga
Yasuyuki Hirano
Affiliation:
Department of Mathematics, Okayamna University, Okayama 700, Japan
Hisao Tominaga
Affiliation:
Department of Mathematics, Okayamna University, Okayama 700, Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent.: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x3 = x. (2) If R is a PI-ring in which every element is the sum of two idempotents, then R/P satisfies the identity x3 = x. (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If RRR is quasi-projective, then R is a finite direct sum of copies of GF(2) and/or GF(3).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 2 , April 1988 , pp. 161 - 164
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]DeMeyer, F. and Ingraham, E., ‘Separable algebras over commutative rings’, in Lecture Note In Math. 181 (Springer-Verlag, Berlin-Heidelberg-New York, 1971).Google Scholar
[2]Miyashita, Y., ‘Quasi-projective modules, perfect modules, and a theorem for modular lattices’, J. Fac. Sci. Hokkaido Univ. Ser I 19 (1966), 86–110.Google Scholar
[3]Rowen, L., ‘Some results on the center of a ring with polynomial identity’, Bull. Amer. Math. Soc. 79 (1973), 219–223.Google Scholar
[4]Schein, B.M., ‘O-rings and LA-rings’, Izv. Vysš. Učeb. Zaved. Matematika 2(51) (1966), 111–122. Amer. Math. Soc. Traimsl. (2) 96 (1970), 137–152.Google Scholar
[5]Tomninaga, H., ‘On a theorem of M. Yamada’, Proc. 9th Sympos. Semigroups and Related Topics (1985), 40–42.Google Scholar