Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T02:32:29.015Z Has data issue: false hasContentIssue false

USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

Published online by Cambridge University Press:  01 April 2014

V. A. HIREMATH
Affiliation:
Department of Mathematics, Karnatak University, Dharwad 580003, India email [email protected]
SHARAD HEGDE*
Affiliation:
Department of Mathematics, Karnatak University, Dharwad 580003, India
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.

In this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Alkan, M., Nicholson, W. K. and Özcan, A. Ç., ‘Strong lifting splits’, J. Pure Appl. Algebra 215 (2011), 18791888.Google Scholar
Borooah, G., Diesl, A. J. and Dorsey, T. J., ‘Strongly clean triangular matrix rings over local rings’, J. Algebra 312 (2007), 773797.Google Scholar
Chen, H., ‘On uniquely clean rings’, Comm. Algebra 39 (2011), 189198.CrossRefGoogle Scholar
Chen, H., ‘Rings with many idempotents’, Int. J. Math. Math. Sci. 22 (3) (1999), 547558.Google Scholar
Lam, T. Y., A First Course in Noncommutative Rings (Springer, New York, 2001).Google Scholar
Nicholson, W. K., ‘Lifting idempotents and exchange rings’, Trans. Amer. Math. Soc. 229 (1977), 269278.Google Scholar
Nicholson, W. K. and Zhou, Y., ‘Rings in which elements are uniquely the sum of an idempotent and a unit’, Glasg. Math. J. 46 (2004), 227236.Google Scholar
Vaserstein, L. N., ‘Bass’s first stable range condition’, J. Pure Appl. Algebra 34 (1984), 319330.Google Scholar